**Differential geometry** studies invariant properties of geometric structures on smooth manifolds,
such as curvature at and distance between points, length of and angle between vectors,
and volume of the space, if such concepts are well-defined on the endowed geometric structure.

**Scalar product** $\langle \cdot, \cdot \rangle$ on a finite-dimensional vector space
is a nondegenerate, symmetric (covariant) 2-tensor:
$q \in \Sigma^2(V^∗)$, $\forall v \in V$, $q(v, V) = 0 \iff v = 0$.
Note that covariant 2-tensors and bilinear forms are identical.
**Scalar product space** $(V, (+, \cdot_\mathbb{F}), \langle \cdot, \cdot \rangle)$
is a finite-dimensional vector space endowed with a specific scalar product.
**Norm** $|v|$ of a vector in a scalar product space is defined as
the square root of the absolute value of its scalar product with itself:
$|v| = \sqrt{|\langle v, v \rangle|}$.
The norm of a scalar product space is not necessarily a norm in the sense of normed space.
**Orthogonal vectors** $v \perp w$ of a scalar product space
are two vectors whose scalar product is zero: $\langle v, w \rangle = 0$.
**Nondegenerate subspace** of a scalar product space is a subspace such that
the restriction of the scalar product to the square of the subspace is nondegenerate.
**Nondegenerate vectors** in a scalar product space is a tuple of vectors
whose principal subtuples span to nondegenerate subspaces.
Every nondegenerate k-tuple in an n-dimensional scalar product space, $k < n$,
can be completed into a nondegenerate basis.
Every orthonormal basis is nondegenerate.
*Gram–Schmidt Algorithm for Scalar Products*:
An orthonormal basis of a scalar product space can be contructed from
a nondegenerate basis such that their principal subtuples of the same size span to the same subspace.
Any scalar product on an n-dimensional vector space
has a canonical matrix representation in some basis: $\text{diag}\{1_r, -1_s\}$, where $r + s = n$.
*Sylvester’s Law of Inertia*:
The canonical matrix representation $\text{diag}\{1_r, -1_s\}$
of a scalar product on a finite-dimensional vector space is unique,
where $r$ is the maximum dimension among all subspaces on which
the restriction of the scalar product is positive definite.
**Index** $s(q)$ of a scalar product on an n-dimensional vector space
is the number of negative terms in its canonical form $\text{diag}\{1_r, -1_s\}$.
**Signature** $(n-s, s)$ of a scalar product
is the tuple of the numbers of its positive terms and negative terms.

**Pseudo-Riemannian metric** (伪黎曼度量) or **semi-Riemannian metric** $g$ on a smooth manifold
is a smooth symmetric 2-tensor field that is nondegenerate and has the same index everywhere:
$g \in \Gamma(\Sigma^2 T^* M)$, $\exists s \in n+1$, $\forall p \in M$, $s(g_p) = s$.
Any Riemannian metric is a pseudo-Riemannian metric.
**Pseudo-Riemannian manifold** is a smooth manifold with a Pseudo-Riemannian metric.
**Einstein metric** is a pseudo-Riemannian metric
whose Ricci tensor satisfies the (mathematicians') Einstein equation $R c = \lambda g$,
where is $\lambda$ is a constant.
Every pseudo-Riemannian manifold admits a collection of smooth orthonormal frames
whose domains cover the manifold.
We use $O(M)$ to denote the set of all orthonormal bases for the tangent spaces on a manifold.

Model pseudo-Riemannian manifolds are
pseudo-Euclidean spaces, pseudo-spheres, and pseudo-hyperbolic spaces.
**Pseudo-Euclidean space** $\mathbb{R}^{r,s}$ of signature $(r, s)$
is the manifold $\mathbb{R}^{r+s}$ with the pseudo-Riemannian metric
$\bar{q}^{(r,s)} = \sum_{i=1}^r (d \xi^i)^2 - \sum_{j=1}^s (d \tau^j)^2$,
where $((\xi^i)_{i=1}^r, (\tau^j)_{j=1}^s)$ is its standard coordinates.
**Pseudo-sphere**...
**Pseudo-hyperbolic space**...

**Lorentz metric** is a pseudo-Riemannian metric of index 1, and thus signature $(r, 1)$.
A smooth manifold admits a Lorentz metric if and only if it admits a rank-1 tangent distribution.
Every noncompact connected smooth manifold admits a Lorentz metric.
A compact connected smooth manifold admits a Lorentz metric if and only if
its Euler characteristic is zero.
**Lorentz manifold** is a smooth manifold with a Lorentz metric.
In the general theory of relativity, spacetime is a Lorentz 4-manifold
whose Ricci curvature satisfies the Einstein field equation $R c - S g / 2 = T$,
where the stress-energy tensor field $T$ describes
density, momentum, and stress of the matter and energy.

Model Lorentz manifolds: the Minkowski spaces, de Sitter spaces, and anti-de Sitter spaces.
Minkowski metric...
**Minkowski space** $\mathbb{R}^{r,1}$ of dimension $r+1$
is the pseudo-Euclidean space of signature $(r, 1)$.
In the special theory of relativity, if gravity is ignored,
spacetime is the 4-dimensional Minkowski space $\mathbb{R}^{3,1}$,
and the laws of physics have the same form in every coordinate system
where the Minkowski metric has the standard expression.

*Fundamental theorem of (pseudo-)Riemannian geometry*:
For every (pseudo-)Riemannian manifold,
there is a unique symmetric connection
on its tangent bundle that is compatible with the (pseudo-)Riemannian metric.
**Levi-Civita connection** $\nabla_g$ on the tangent bundle of a (pseudo-)Riemannian manifold
is the symmetric connection that is compatible with the (pseudo-)Riemannian metric.
For objects determined by the (pseudo-)Riemannian metric of a (pseudo-)Riemannian manifold,
whenever there is no ambiguity about the metric, we can remove such explicit dependence in notations,
e.g. use $\nabla$ to denote the Levi-Civita connection.
The Levi-Civita connection on a (pseudo-)Euclidean space is the Euclidean connection.
The Levi-Civita connection on an embedded (pseudo-)Riemannian submanifold
of a (pseudo-)Euclidean space is the tangential connection.
**Christoffel symbols** $\Gamma_{ij}^k$ of a (pseudo-)Riemannian metric
are the connection coefficients of the Levi-Civita connection in a smooth coordinate chart:
$\Gamma_{ij}^k = \frac{1}{2} g^{kl} (\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij})$.

**Koszul's formula** for the inner product of a covariant derivative
w.r.t. the Levi-Civita connection and a smooth vector field:
$\langle \nabla_X Y, Z \rangle = \frac{1}{2} \Big(
X \langle Y, Z \rangle + Y \langle Z, X \rangle - Z \langle X, Y \rangle
+\langle Y, [Z, X] \rangle + \langle Z, [X, Y]\rangle - \langle X, [Y, Z] \rangle \Big)$.
Given coefficients $c_{ij}^k$ of the Lie bracket in a smooth local frame,
such that $[E_i, E_j] = c_{ij}^k E_k$,
the coefficients of the Levi-Civita connection in this frame are:
$\Gamma_{ij}^k = \frac{1}{2} g^{kl} (E_i g_{jl} + E_j g_{il} - E_l g_{ij} -
g_{jm} c_{il}^m - g_{lm} c_{ji}^m + g_{im} c_{lj}^m)$.
If the frame is orthonormal, this is simplified as
$\Gamma_{ij}^k = \frac{1}{2} (c_{ij}^k - c_{ik}^j - c_{jk}^i)$.

**Exponential map** (指数映射) $\exp: \mathscr{E} \mapsto M$ of a (pseudo-)Riemannian manifold
maps every tangent vector to the point on its geodesic at time one, if it is defined:
$\exp(v) = \gamma_v(1)$;
its domain $\mathscr{E}$ is the subset of the tangent bundle $T M$ such that
the maximal geodesic $\gamma_v$ is defined on an interval containing $[0,1]$.
**Restricted exponential map** $\exp_p$ at a point on the manifold
is the restriction of the exponential map to the tangent space at the point:
$\exp_p = \exp|_{\mathscr{E} \cap T_pM}$.
The exponential map of a (pseudo-)Riemannian manifold:
(1) is defined on an open subset of its tangent bundle,
which is star-shaped when restricted to each tangent space;
(2) is a smooth map, whose differential at each $(p, 0)$ is the identity map of tangent space $T_p M$;
(3) is a diffeomorphism between a neighborhood of each $(p, 0)$ on tangent space $T_p M$
and a neighborhood of the point on the manifold;
(4) encodes all the maximal geodesics: $\exp(t v) = \gamma_v(t)$.
A (pseudo-)Riemannian manifold is geodesically complete if and only if
its exponential map is defined for the entire tangent bundle: $\exp: T M \mapsto M$.
Geodesically complete manifolds are the natural setting for
global questions in (pseudo-)Riemannian geometry.
Every Euclidean space is geodesically complete.
No proper open subset of a Euclidean space
with its Euclidean metric or with a pseudo-Euclidean metric is geodesically complete.

**Normal neighborhood** of a point in a (pseudo-)Riemannian n-manifold
is the image of a star-shaped neighborhood of zero of the tangent space
under the restricted exponential map where it is a diffeomorphism:
$U \subset M$: $\exists V \subset T_p M$, $V = \bigcup_{a \in [0, 1]} a V$, and $\exp_p: V \cong U$.
Note that this is not the normal neighborhood of an embedded submanifold.
(Pseudo-)Riemannian **normal coordinates** $\varphi: U \mapsto \mathbb{R^n}$
centered at a point in a (pseudo-)Riemannian n-manifold
is a smooth coordinate chart on a normal neighborhood of the point
that maps a point to the coordinate representation of its preimage under the exponential map,
w.r.t. an orthonormal basis for the tangent space:
$\varphi^{-1}(x^i)_{i = 1}^n = \exp_p(x^i b_i)$, where
$(b_i, b_j) = \delta_{ij}$, $T_p M = \oplus_{i = 1}^n b_i$, $x^i b_i \in V$, $\exp_p: V \cong U$.
For every orthonormal basis for the tangent space,
there is a unique normal coordinate chart on the normal neighborhood
whose coordinate vectors at that point match the orthonormal basis.
In any normal coordinate chart,
the Riemannian metric at the point is represented by the Kronecker delta,
and the geodesics starting at the point are represented by straight lines (aka radial geodesics):
$g_{ij} = \delta_{ij}$, $\gamma_v (t) = (t v^i)_{i=1}^n \subset U$.

Figure: Curvature concepts.

**Riemann curvature endomorphism** or **(1,3)-curvature tensor** $R$ on a (pseudo-)Riemannian manifold
is the (1,3)-tensor field identified with the multilinear map
$R: \mathfrak{X}^3(M) \mapsto \mathfrak{X}(M)$, $\forall X, Y, Z \in \mathfrak{X}(M)$,
$R(X, Y) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$,
where $[\cdot,\cdot]$ is the Lie bracket of smooth vector fields.
Its coordinate representation in a smooth chart is:
$R = R_{ijk}^l d x^i \otimes d x^j \otimes d x^k \otimes \partial_l$,
where the coeffients satisfy $R(\partial_i, \partial_j) \partial_k = R_{ijk}^l \partial_l$,
and can be computed as $R_{ijk}^l = \partial_i \Gamma_{jk}^l - \partial_j \Gamma_{ik}^l +
\Gamma_{jk}^p \Gamma_{ip}^l - \Gamma_{ik}^p \Gamma_{jp}^l$.
**Curvature endomorphism** $R(X, Y)$
determined by two smooth vector fields on a (pseudo-)Riemannian manifold
is the smooth bundle endomorphism of the tangent bundle
$R(X, Y): \mathfrak{X}(M) \mapsto \mathfrak{X}(M)$.
*Ricci Identities* (all except (1)):
Curvature endomorphism measures the failure of second covariant derivatives to commute:
(1) covariant derivative along families of curves:
$\Gamma \in C^\infty(J \times I, M)$, $D_s D_t - D_t D_s = R(\partial_s \Gamma, \partial_t \Gamma)$;
(2) total covariant derivative of smooth vector field:
$\nabla_{X,Y}^2 - \nabla_{X,Y}^2 = R(X, Y)$;
(3) total covariant derivative of smooth covector field:
$\nabla_{X,Y}^2 - \nabla_{X,Y}^2 = -R(X, Y)^∗$;
(4) total covariant derivative of smooth (k,l)-tensor field:
$\nabla_{X,Y}^2 B - \nabla_{X,Y}^2 B = \sum_{i \in k} B \circ_i R(X,Y)^∗ -
\sum_{j \in l} B \circ_{k+j} R(X,Y)$,
where $\circ_i$ denotes composition at the i-th argument, that is,
$(f \circ_i g)(x_j)_{j=1}^n = f(y_j)_{j=1}^n$, $y_i = g(x_i)$, $\forall j \ne i$, $y_j = x_j$.

**Riemann curvature tensor** $Rm$ or $Riem$ on a (pseudo-)Riemannian manifold
is the covariant 4-tensor field obtained from the curvature endomorphism by lowering an index:
$Rm = R^\flat$; that is, $Rm(X, Y, Z, W) = \langle R(X, Y) Z, W \rangle_g$;
in a smooth chart, $Rm = R_{ijkl} d x^i \otimes d x^j \otimes d x^k \otimes d x^l$,
where $R_{ijkl} = g_{lm} R_{ijk}^m = g_{lm} (\partial_i \Gamma_{jk}^m - \partial_j \Gamma_{ik}^m +
\Gamma_{jk}^p \Gamma_{ip}^m - \Gamma_{ik}^p \Gamma_{jp}^m)$.
The curvature tensor of a (pseudo-)Riemannian manifold is a local isometry invariant.
The curvature tensor has the following symmetries,
where 1, 2, 3, 4 denotes vectors fields for simplicity:
(1) $Rm(1, 2, 3, 4) = - Rm(2, 1, 3, 4)$;
(2) $Rm(1, 2, 3, 4) = - Rm(1, 2, 4, 3)$;
(3) $Rm(1, 2, 3, 4) = Rm(3, 4, 1, 2)$;
(4) $Rm(1, 2, 3, 4) + Rm(2, 3, 1, 4) + Rm(3, 1, 2, 4) = 0$.
Written in components w.r.t a basis:
(1) $R_{ijkl} = -R_{jikl}$;
(2) $R_{ijkl} = -R_{ijlk}$;
(3) $R_{ijkl} = R_{klij}$;
(4) $R_{ijkl} + R_{jkil} + R_{kijl} = 0$.
For (pseudo-)Riemannian 1-manifolds, the curvature tensor is zero.
For (pseudo-)Riemannian 2-manifolds, the curvature tensor at each point can be specified by a scalar:
in any basis, $R_{0101} = R_{1010} = -a$, $R_{1001} = R_{0110} = a$, $R_{ijkl} = 0$ otherwise;
the scalar is related to the scalar curvature $S = 2a \det(g_{ij})$;
in any orthonormal basis, this scalar equals the Gaussian curvature: $a = S/2 = K$.
**Algebraic curvature tensor** on a vector space
is a covariant 4-tensor that have the symmetries of a Riemann curvature tensor.
The vector space of algebraic curvature tensors on a vector space is denoted as $\mathcal{R}(V^∗)$.

Contracting the curvature tensor gives two more curvature concepts.
**Ricci curvature** or **Ricci tensor** $Rc$ or $Ric$ on a (pseudo-)Riemannian manifold
is the symmetric 2-tensor field defined as
the trace of the curvature endomorphism on its first and last indices:
$Rc(X, Y) = \text{tr}(R(\cdot, X) Y)$, where $X$ and $Y$ are vector fields;
written in components, $R_{ij} := R_{kij}^k = g^{km} R_{kijm}$.
**Trace** $\text{tr}_g h$ of a covariant k-tensor field on a Riemannian manifold, $k \ge 2$,
w.r.t. the Riemannian metric, is the covariant (k-2)-tensor field defined by the trace of
the (1,k-1)-tensor field obtained by raising its last index: $\text{tr}_g h = \text{tr}(h^\sharp)$;
for covariant 2-tensor fields, in a local frame, $\text{tr}_g h = h_i^i = g^{ij} h_{ij}$.
**Scalar curvature** $S(p)$ on a (pseudo-)Riemannian manifold
is the real-valued function on the manifold defined as the trace of the Ricci tensor:
$S := \text{tr}_g Rc = R_i^i = g^{ij} R_{ij}$.
Since the curvature tensor is a local isometry invariant,
so are the Ricci curvature and the scalar curvature.
Global conclusions on (pseudo-)Riemannian manifolds
often assume an upper or lower bound for the sectional curvature.
Except when $n < 4$, assuming a bound on Ricci or scalar curvature
is a strictly weaker hypothesis than assuming one on sectional curvature.
When $n \ge 4$, a bound on the Ricci curvature implies nothing about sectional curvatures;
for example, Calabi–Yau manifolds are compact Riemannian manifolds
that have zero Ricci curvature but nonzero sectional curvatures.
And the problem of drawing global conclusions from scalar curvature bounds is far more subtle.
Lower bounds on Ricci curvature are often specified as:
$\forall v \in T M$, $|v| = 1$, $\exists c \in \mathbb{R}$: $Rc(v, v) \ge (n-1) c$;
because Ricci curvature on a unit vector is a sum of (n-1) sectional curvatures,
bounds on Ricci curvature are often multiplied by (n-1) for ease of comparison;
upper bounds are used less often.

*Ricci Decomposition of the Curvature Tensor*:
For a (pseudo-)Riemannian n-manifold, $n \ge 3$,
with direct sum decomposition $\mathcal{R}(V^∗) = \ker(\text{tr}_g) \oplus \ker(\text{tr}_g)^\perp$,
curvature tensor is orthogonally decomposed as $Rm = W + P \mathbin{\bigcirc\mspace{-15mu}\wedge} g$,
where $W$ is the Weyl tensor, and Schouten tensor is orthogonally decomposed as
$P = \frac{1}{n-2} \mathring{Rc} + \frac{1}{2n(n-1)} S g$,
because the traceless Ricci tensor $\mathring{Rc} = Rc - S g / n$
satisfies $\langle\mathring{Rc}, g\rangle = 0$.
When $n=3$, the Weyl tensor is zero, and thus
the curvature tensor is determined by the Ricci curvature.
When $n=2$, the traceless Ricci tensor is zero, and thus
the Ricci tensor and the curvature tensor is determined by the scalar curvature.

Roughly speaking, positive curvature causes nearby geodesics to converge, while negative curvature causes them to spread out. Jacobi field provides a quantitative way to measure the effect of curvature on a one-parameter family of geodesics.

**One-parameter family of curves** in a smooth manifold
is a continuous map from a rectangle to the manifold:
$\Gamma \in C^0(J \times I, M)$, where I and J are intervals.
It defines two collections of curves:
**main curves** $\Gamma_s(t) = \Gamma(s,t)$;
**transverse curves** $\Gamma^{(t)}(s) = \Gamma(s,t)$.
The family is called **admissible** if:
(1) J is open and I is closed;
(2) every main curve is admissible;
(3) Γ can be partitioned along I (i.e. an admissible partition) into smooth maps.
**Variation of an admissible curve** is an admissible family of curves
such that J contains 0 and the 0th main curve matches the curve: $\Gamma_0 = \gamma$.
**Proper variation** of an admissible curve is one
where all the main curves have the same starting and ending points.
**Variation field** of a variation of an admissible curve
is its transverse vector field along the curve:
$V(t) = \partial_s \Gamma(0,t)$
The variation field of a variation is a piecewise smooth vector field along the curve.
**Proper vector field** along a curve is a vector field that vanishes at the end points.
The variation field of every proper variation is proper along the curve.

**Variation through geodesics** is a variation of a geodesic
where every main curve is a geodesic.
Note that variations of geodesics arise in, for example,
geodesic regression and curve fitting on manifolds
(given a family of curves $\gamma: \mathbb{R} \mapsto M$,
find one that minimizes a balance between total acceleration and distance to points).
A vector field J along a geodesic γ in a (pseudo-)Riemannian manifold
is the variation field of a variation through geodesics,
only if it satisfies the **Jacobi equation**: $D_t^2 J + R(J, \gamma') \gamma' = 0$,
where R is the (1,3)-curvature tensor and D is the covariant derivative.
**Jacobi field** along a geodesic in a (pseudo-)Riemannian manifold
is a smooth vector field along the geodesic that satisfies the Jacobi equation.
The set of Jacobi fields along a geodesic γ is denoted as $\mathfrak{J}(\gamma)$,
which is a 2n-dimensional linear subspace of $\mathfrak{X}(\gamma)$.
**Initial value problem for Jacobi fields**:
given a geodesic $\gamma(t) = \exp_p(t v)$
and two tangent vectors (p, v') and (p, a),
find a Jacobi field J along γ such that J(0) = v' and D_t J(0) = a.
Note that this problem can be specified by (p, v, v', a).
*Existence and uniqueness of Jacobi fields*:
Every initial value problem (p, v, v', a) has a unique solution.
**Two-point boundary problem for Jacobi fields**:
given a geodesic $\gamma(t) = \exp_p(t v)$
and two tangent vectors (p, v') and (q, w), where $q = \exp_p(v)$,
find a Jacobi field J along γ such that J(0) = v' and J(1) = w.
Note that this problem can be specified by (p, v, v', w).
The two-point boundary problem (p, v, v', w) has a unique solution
if and only if (p, v) is not a tangent conjugate point (see definition below).

**Conjugate points** along a geodesic γ in a (pseudo-)Riemannian manifold is a pair of points in γ
such that there is a Jacobi field along γ that vanishes at them but not everywhere.
Intuitively, if two points are conjugate along a geodesic,
one expects to find a one-parameter family of geodesic segments
that start at one and end (almost) at the other.
The following fact allows for an equivalent definition of conjugate points
that does not use geodesic or Jacobi field:
A tangent vector (p, v) to a (pseudo-)Riemannian manifold is a critical point of $\exp_p$
if and only if $\exp_p(v)$ is conjugate to p along $\gamma(t) = \exp_p(t v)$.
Tangent **conjugate point** of a (pseudo-)Riemannian n-manifold
is a tangent vector (p, v) that is a critical point of the restricted exponential map:
$d(\exp_p)_v T_p M \subsetneq T_q M$, where $q = \exp_p(v)$.
Tangent **conjugate locus** $\text{Conj}(p)$ of a point p in a (pseudo-)Riemannian manifold
is the set of tangent conjugate points in its tangent space:
$\text{Conj}(p) = \{v \in T_p M : \text{rank}~d(\exp_p)_v < n\}$.
For the unit n-sphere (n ≥ 2) and the unit projective n-space (n ≥ 3),
the tangent conjugate locus of any point consists of
all (n-1)-spheres with radius a multiple of π:
$\forall p \in \mathbb{S}^n$, $\text{Conj}(p) = \cup_{k \in \mathbb{N}} k \pi \mathbb{S}^{n-1}$.
Note that, for the lower dimensional cases excluded above, they have no conjugate point.
Properties of cut points and conjugate points [@Lee2018, Prop 10.20, 10.32]:
given a tangent vector (p, v),
$t_1 v$ is the (tangent) cut point
if and only if $\gamma(t) = \exp_p(t v)$ stops minimizing at $t_1$;
$t_2 v$ is a tangent conjugate point
if and only if $\exp_p$ fails to be a local diffeomorphism at $t_2 v$;
we have $0 < t_1 \le t_2 \le \infty$.

Figure: Visualization of a tangent space, by pulling back via the exponential map: (left) the unit 2-sphere; (right) the unit projective plane. The tangent cut locus is the circle of radius π for $\mathbb{S}^2$, and of radius π/2 for $\mathbb{P}^2$; any ray beyond its cut point can be found within the ball to the cut point. The tangent conjugate locus consists of circles of radius n π (n = 1, 2, ...), for both cases; around every tangent conjugate point, some point becomes a submanifold. Note that for $\mathbb{S}^2$, TCL(p) ⊂ Conj(p); while for $\mathbb{P}^2$, the two do not intersect.

Riemannian geometry (per Bernhard Riemann) is a generalization of the intrinsic geometry of surfaces in the three-dimensional Euclidean space (see C.F. Gauss), and is concerned with invariant properties of Riemannian manifolds under isometries.

**Riemannian metric** (黎曼度量) $g$ on a smooth manifold
is a smooth symmetric 2-tensor field whose value at each point is an inner product:
$g \in \Gamma(\Sigma^2 T^* M)$, $\forall (p, v) \in T M$, $v \ne 0 \implies g_p(v, v) > 0$.
A Riemannian metric defines an inner product $\langle \cdot, \cdot \rangle_g$
on each tangent space of the manifold: $\langle v, w \rangle_g = g_p(v, w)$.
Every smooth manifold admits a Riemannian metric.
**Riemannian manifold** $(M, g)$ is a smooth manifold endowed with a Riemannian metric.
Different Riemannian metrics on the same manifold can have vastly different geometric properties,
see [@Lee2012, Problem 13-20].
**Euclidean metric** $\bar g$ on a Euclidean space
is the Riemannian metric that equals the Euclidean inner product everywhere:
$\bar g_p (v, w) = v \cdot w$.
**Euclidean space** $(\mathbb{R}^n, \mathcal{T}, \mathcal{A}, \bar g)$
is the Riemannian manifold consisting of the smooth manifold and the Euclidean metric
associated with a Euclidean space (as an inner product space
$(\mathbb{R}^n, +, \cdot_{\mathbb{R}}, (\cdot,\cdot))$).
Because the Euclidean space as a Riemannian manifold
is unambiguously constructed from the Euclidean space as an inner product space,
there is no need to distinguish these two terms.
Any n-dimensional real inner product space is isometric with
the Euclidean n-space as Riemannian manifolds: $V \cong \mathbb{R}^n$.

**Local coordinate representation** $(g_{ij})$ of a Riemannian metric w.r.t. a smooth frame
is the smooth function whose components are the inner products of the coordinate frames:
$g_{ij} = \langle e_i, e_j \rangle_g$.
Given a smooth frame, a Riemannian metric can be written uniquely as
a linear combination of the tensor products of the dual covectors:
$g = g_{ij} \varepsilon^i \otimes \varepsilon^j$.
Because all covariant 1-tensors are symmetric,
and local coordinate representations of Riemannian metrics are also symmetric,
the previous formula can be written in terms of symmetric products of dual covectors:
$g = g_{ij} \varepsilon^i \varepsilon^j$, where $\varepsilon^i \varepsilon^j =
(\varepsilon^i \otimes \varepsilon^j + \varepsilon^j \otimes \varepsilon^i) / 2$.
Given a smooth chart, the previous formula becomes $g = g_{ij} d x^i d x^j$, where
$g_{ij} = \left\langle \frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j} \right\rangle_g$.
The action $\langle X, Y \rangle_g$ of a Riemannian metric on smooth vector fields
is a smooth real-valued function.
Given an smooth frame, the action can be written as $\langle X, Y \rangle_g = g_{ij} X^i Y^j$.
Vector fields on a Riemannian manifold are **orthonormal**
if they are orthonormal w.r.t. the inner product at each point.
**Orthonormal frame** on an open subset of a Riemannian manifold
is a frame consisting of orthonormal vector fields: $\langle e_i, e_j \rangle_g = \delta_{ij}$.
Every Riemannian manifold admits a collection of smooth orthonormal frames
whose domains cover the manifold.
The local coordinate representation of the Riemannian metric of a Riemannian manifold
in any orthonormal frame is the Kronecker delta: $g_{ij} = \delta_{ij}$.
The standard global coordinate frame of a Euclidean space is orthonormal,
i.e. the standard coordinate representation of the Euclidean metric is the Kronecker delta:
$\bar g_{ij} = \delta_{ij}$.
**Adapted frame** to a smooth n-submanifold of a Riemannian m-manifold
is a frame whose first $n$ component vector fields are tangent to the submanifold.
For an embedded submanifold of a Riemannian manifold,
there are smooth orthonormal frames that are adapted to and cover the submanifold.

Riemannian **isometry** $F: M_1 \mapsto M_2$ between Riemannian manifolds is a diffeomorphism
by which the pullback of the Riemannian metric of its codomain equals that of its domain:
$F^∗ g_2 = g_1$.
Equivalently, an isometry is a diffeomorphism whose pushforward at each point is a linear isometry.
**Isometry group** $\text{Iso}(M, g)$ on a Riemannian manifold
is the group consisting of the set of all isometries on the manifold and composition.
*Myers–Steenrod theorem*:
The isometry group on a Riemannian manifold is a Lie group acting smoothly on the manifold.
**Isometric immersion** / **isometric embedding** / **local isometry** between Riemannian manifolds
is a smooth immersion / smooth embedding / local diffeomorphism defined analogously to isometry.

**Norm** or **length** $|v|_g$ of a tangent vector at a point in a Riemannian manifold
is the square root of the inner product of the vector and itself:
$|v|_g = \sqrt{\langle v, v \rangle_g}$.
**Angle** between two nonzero tangent vectors at a point in a Riemannian manifold is:
$\theta = \arccos \left(\frac{\langle v, w \rangle_g}{|v|_g |w|_g}\right)$.
Norm and angle (as smooth tensor fields) on a Riemannian manifold are local isometry invariants.
Open **geodesic ball** $\exp_p(B_\varepsilon(0))$ of radius ε at a point in a Riemannian manifold
is the diffeomorphic image of the ε-ball of a tangent space under the restricted exponential map:
$\exp_p: B_\varepsilon(0) \cong \exp_p(B_\varepsilon(0))$.
Note that the radius of the geodesic ball is measured w.r.t. the norm on the tangent space.
By definition, an open geodesic ball is a normal neighborhood.
**Closed geodesic ball** $\exp_p(\bar{B}_\varepsilon(0))$ in a Riemannian manifold
is the diffeomorphic image of a closed ε-ball in an open set of a tangent space
under the restricted exponential map:
$\bar{B}_\varepsilon(0) \subset V \in \mathcal{T}(T_p M)$, $\exp_p: V \cong \exp_p(V)$.
**Geodesic sphere** $\exp_p(\partial B_\varepsilon(0))$ is the boundary of a closed geodesic ball.
**Uniformly δ-normal neighborhood** of a point in a Riemannian manifold is a normal neighborhood
such that every point in it has a geodesic δ-ball and it is contained in all of them:
$U \subset M$: $\exists \delta > 0$, $\forall p \in U$, $U \subset \exp_p(B_\delta(0))$.
**Injectivity radius** $\text{inj}(p)$ of a Riemannian manifold at a (non-boundary) point
is the supremum of the radii of its geodesic balls:
$\text{inj}(p) = \sup\{\varepsilon > 0 : \exp_p(B_\varepsilon(0))\}$.
Injectivity radius over a complete, connected Riemannian manifold is continuous:
$\text{inj} \in C^0(M, \mathbb{R}_{>0})$.
**Injectivity radius** $\text{inj}(M)$ of a Riemannian manifold without boundary
is the infimum of the injectivity radii at all points:
$\text{inj}(M) = \inf\{\text{inj}(p) : p \in M\}$.
Every compact Riemannian manifold without boundary has a positive injectivity radius.

**Speed** $|\gamma'(t)|$ or $|\dot \gamma(t)|$ of a smooth curve in a Riemannian manifold
at a time instance is the norm of its velocity at that time.
**Constant-speed curve** in a Riemannian manifold
is a piecewise smooth curve whose speed is constant wherever it is smooth.
**Unit-speed curve** in a Riemannian manifold
is a piecewise regular curve whose speed is one wherever it is smooth.
A piecewise regular curve is **unit-speed parameterized** if it is unit-speed.
Every regular curve in a Riemannian manifold has a unit-speed forward reparameterization.
**Length** $L_g(\gamma)$ of a piecewise regular curve segment in a Riemannian manifold
is the definite integral of the norm of its velocity: $L_g(\gamma) = \int_a^b |\gamma'(t)|_g dt$.
Length is independent of the parameterization of the curve.
Length of piecewise regular curve segments in a Riemannian manifold is a local isometry invariant.
The length of a unit-time geodesic segment in a Riemannian manifold equals its velocity:
$L_g(\gamma_v|_{[0,1]}) = v$.
**Arc-length function** $s(t)$ of a piecewise regular curve segment in a Riemannian manifold
is the integral of the norm of its velocity with variable upper limit:
$s(t) = \int_a^t |\gamma'(t)|_g dt$.
The arc-length function of a geodesic $\gamma_v$ in a Riemannian manifold is $s(t) = v t$.
A piecewise regular curve segement is **parameterized by arc length**
if it is unit-speed and its domain starts from zero.
Every piecewise regular curve segement in a Riemannian manifold
has a unique forward reparameterization by arc length.

**Minimizing curve** in a Riemannian manifold is a curve with the smallest length
among all admissible curves (piecewise regular curve segment) with the same endpoints:
$L_g(\gamma) = \min\{L_g(\tilde \gamma) : \tilde\gamma(a)=\gamma(a), \tilde\gamma(b)=\gamma(b)\}$.
Every minimizing curve with a unit-speed parameterization is a geodesic.
**Locally minimizing** curve in a Riemannian manifold is a piecewise regular curve
such that every point has a neighborhood in which every curve segment is minimizing:
$\forall t_0 \in I$, $\exists I_0 \subset I$: $t_0 \in I_0$, $\forall [a, b] \subset I_0$,
$L_g(\gamma|_{[a,b]}) = \min\{L_g(\tilde \gamma) : \tilde\gamma(a)=\gamma(a),
\tilde\gamma(b)=\gamma(b)\}$.
Every geodesic in a Riemannian manifold is locally minimizing.

**Riemannian distance** $d_g(p, g)$ between two points in a connected Riemannian manifold
is the infimum of the lengths of all piecewise regular curve segments between the points:
$d_g(p, g) = \inf \{L_g(\gamma) : \gamma(a) = p, \gamma(b) = q\}$.
**Riemannian distance function** $d_g$ of a connected Riemannian manifold
is the function that provides the distance between any pair of points in the manifold.
The Riemannian distance function of a Riemannian manifold is an isometry invariant.
A connected Riemannian manifold together with the Riemannian distance function
is a metric space $(M, d_g)$ and the metric topology $\mathcal{T}_{d_g}$
is the same as the manifold topology.
**Bounded Riemannian manifold** is a connected Riemannian manifold with a finite diameter:
$\sup_{x,y \in M} d_g(x, y) < \infty$.
Metrically **complete Riemannian manifold** is a connected Riemannian manifold such that
the metric space it generates is complete, i.e. every Cauchy sequence converges.
*Hopf–Rinow Theorem*: A connected Riemannian manifold is metrically complete
if and only if it is geodesically complete.
Thus we can simply say a Riemannian manifold is complete,
which is understood to be both geodesically complete and metrically complete if it is connected,
and is geodesically complete and each component is metrically complete if it is disconnected.
Any two points in a complete, connected Riemannian manifold
can be joined by a minimizing geodesic segment.
**Complete Riemannian metric** is the Riemannian metric of a complete, connected Riemannian manifold.
Every connected smooth manifold admits a complete Riemannian metric.
**Metrizable topological space** is one that admits a distance function
whose metric topology is the same as the given topology.
Every smooth manifold is metrizable.

Note that from any point in a geodesically complete (pseudo-)Riemannian manifold,
the range of the restricted exponential map is a connected component,
which can be pulled back to the tangent space,
where each point may appear countably or uncountably infinitely many times:
$\exp_p: T_p M \mapsto M$, $\exp_p(T_p M) = M_p$, $\exp_p^{-1}(M_p) = T_p M$.
For a connected complete Riemannian manifold, the following definitions are useful.
**Cut time** of a tangent vector is the supremum of time while the geodesic is minimizing:
$t_\text{cut}(p, v) = \sup\{b > 0 : d_g(p, \exp_p(b v)) = |b v|\}$.
**Cut point** of a point along a geodesic with a finte cut time
is the point beyond which the geodesic stops minimizing: $\gamma_v (t_\text{cut}(p, v))$.
**Cut locus** $\text{Cut}(p)$ of a point is the set of all its cut points:
$\text{Cut}(p) = \{p' \in M : \exists v \in T_p M, p' = \gamma_v (t_\text{cut}(p, v))\}$.
**Tangent cut locus** $\text{TCL}(p)$ of a point
is the set of its tangent vectors with a unit cut time:
$\text{TCL}(p) = \{v \in T_p M : t_\text{cut} (p, v) = 1\}$.
**Injectivity domain** $\text{ID}(p)$ of a point
is the set of its tangent vectors with cut time less than one:
$\text{ID}(p) = \{v \in T_p M : t_\text{cut} (p, v / |v|) < 1\}$.
Apparenlty, cut locus is the image of the tangent cut locus,
and tangent cut locus is the boundary of the injectivity domain.
Here is an equivalent set of definitions that do not use cut time, tangent vector, or geodesic:
**injectivity domain** is the largest open subset of the domain
of the restricted exponential map where it is radially isometric,
$\text{ID}(p) = \text{Int}\{v \in T_p M : d_g(p, \exp_p(v)) = |v|\}$;
**tangent cut locus** is the boundary of the injectivity domain,
$\text{TCL}(p) = \partial \text{ID}(p)$;
**cut locus** is the boundary of the image of the injectivity domain:
$\text{Cut}(p) = \partial \exp_p(\text{ID}(p))$,
or equivalently, the image of the tangent cut locus,
$\text{Cut}(p) = \exp_p(\text{TCL}(p))$.
Note that injectivity domain is a star-shaped neighborhood of the tangent space,
and the injectivity radius is the distance to its cut locus
if the cut locus is nonempty, and infinite otherwise.
*Theorem*: For every point in a connected complete Riemannian manifold,
(a) its cut locus is a measure-zero (closed) subset;
(b) the image of the closure of the injectivity domain is the entire manifold,
$\exp_p(\overline{\text{ID}(p)}) = M$;
(c) the image of the injectivity domain is the largest normal neighborhood of the point,
$\exp_p|_{\text{ID}(p)}: \text{ID}(p) \cong M \setminus \text{Cut}(p)$.
We may denote the normal neighborhood
$\hat{U}(p) = M \setminus \text{Cut}(p) = \exp_p(\text{ID}(p))$.
*Corollary*: Every compact, connected, smooth n-manifold is homeomorphic to
a quotient space of the closed n-ball $\bar{\mathbb{B}}^n$,
by an equivalence relation that identifies only points on the boundary.
*Condition for symmetry* [@Sakai1996, Ch. III, Prop. 4.1]:
Two points are in each other's cut locus if there is more than one shortest geodesic between them.
The determination of the cut locus of a point is typically very difficult.
For the unit n-sphere, the cut locus of any point is its antipodal point;
the tangent cut locus of any point is the radius-π (n-1)-sphere,
$\forall p \in \mathbb{S}^n$, $\text{TCL}(p) = \pi \mathbb{S}^{n-1}$;
and the n-sphere is homeomorphic to the closed n-ball by identifying all boundary points to one,
$\mathbb{S}^n \cong \bar{\mathbb{B}}^n / \mathbb{S}^{n-1}$.
For the unit projective n-space (n ≥ 2),
the tangent cut locus of any point is the (n-1)-sphere of radius π/2.
**Riemannian logarithm** $\log_p$ at a point of a connected complete Riemannian manifold
is the inverse of the exponential map on the injectivity domain:
$\log_p: M \setminus \text{Cut}(p) \mapsto \text{ID}(p)$, $\log_p = (\exp_p|_{\text{ID}(p)})^{-1}$.
It pulls back each point in the normal neighborhood to the unique smallest tangent vector.
Note that the Riemannian logarithm is generally not an isometry:
although rays in the injectivity domain correspond to minizing geodesics starting from the point,
line segments in the injectivity domain in general do not correspond to geodesics.
For a 2-sphere, the equator is mapped to a circle,
and other geodesics not passing the point are mapped to curves.
It can be helpful to extend the domain of the Riemannian logarithm to the entire manifold:
**generalized Riemannian logarithm** is the correspondence
$\log_p: M \rightrightarrows \text{ID}(p) \cup \text{TCL}(p)$,
$\log_p = (\exp_p|_{\text{ID}(p) \cup \text{TCL}(p)})^{-1}$.
Note that it restricts to the Riemannian logarithm,
but for a point p' in the cut locus, it gives the non-singleton pre-image
$\exp_p^{-1}(p') \cap \text{TCL}(p)$.

**Geodesically convex subset** of a Riemannian manifold is a subset such that,
between any pair of points in it, the minimizing geodesic segment is unique
and (its image) is included in the subset.
Every sufficiently small geodesic ball in a Riemannian manifold is geodesically convex:
$\forall p \in M$, $\exists \varepsilon_0 > 0$: $\forall \varepsilon \le \varepsilon_0$,
$\exp_p(B_\varepsilon(0))$ is geodesically convex.

Notes: If two Riemannian metrics on a connected smooth manifold determine the same distance function, then the Riemannian metrics are the same. (What kind of distance function admit a Riemmannian metric?) Nonsmooth metrics do not admit a Riemmannian metric, e.g. 1-norm, ∞-norm, and any p-norm for $p \in (0, 1)$. A finite set in a metric space can be isometrically embedded in a Euclidean space ($d(x,y) = |x-y|$) if and only if the kernel $k(x, y) = e^{-d(x,y)}$ is positive semi-definite; the top eigenvectors provide low-dimension embeddings with the smallest distortion in terms of Frobenius norm [@Schoenberg1937]. (How to estimate the Riemmannian metric compatible with a distance function?)

One may construct Riemannian manifolds from others, typically in three ways: Riemannian submanifold, Riemannian submersion, or product manifold.

**Pullback metric** $F^∗ g$ on a smooth manifold by a smooth immersion $F: M \mapsto N$
to a Riemannian manifold $(N, g)$ is the pullback of the Riemannian metric by the immersion.
The pullback of a Riemannian metric by a map is a Riemannian metric on the domain
if and only if the map is a smooth immersion.
**Induced metric** $\iota^∗ \tilde{g}$,
previously known as **first fundamental form** (as in bilinear form),
on a smooth submanifold $M$ of a Riemannian manifold $(\tilde{M}, \tilde{g})$
is the pullback of the Riemannian metric by the inclusion map, or equivalently,
the restriction of the metric to tangent spaces of the submanifold at each point:
$\forall p \in M$, $(\iota^∗ \tilde{g})_p = \tilde{g}_p |_{(T_p M)^2}$.
**Riemannian submanifold** $(M, g)$ of a Riemannian manifold $(\tilde{M}, \tilde{g})$
is a smooth submanifold endowed with the induced metric: $g = \iota^∗ \tilde{g}$.
Note that $M$ can be an immersed or embedded submanifold.
Computation on a (Riemannian) n-submanifold of a Riemannian m-manifold
is usually carried out most conveniently in a smooth local parameterization:
$X \in C^\infty(\varphi(U), \tilde{M})$,
where $U \subset M$, $\varphi \in \mathcal{A}$, $\varphi(U) \subset \mathbb{R}^n$.
The metric induced on the local parameter space
equals the pullback of the Riemannian metric of the ambient manifold:
$\hat{g} = X^∗ g = X^∗ \tilde{g}$.
If the ambient manifold is the Euclidean m-space,
the induced metric on the parameter space has coordinate representation:
$\hat{g} = X^∗ \bar g = \sum_{i=1}^m (d X^i)^2$, which expands to $\sum_{i,j=1}^n \sum_{k=1}^m
\left(\frac{\partial X^k}{\partial x^i} \frac{\partial X^k}{\partial x^j}\right) d x^i d x^j$, i.e.
$\hat g_{ij} = \sum_{k=1}^m \frac{\partial X^k}{\partial x^i} \frac{\partial X^k}{\partial x^j}$.
In particular, the metric induced the graph parameterization $\varphi_f(x) = (x, f(x))$
of the graph $\Gamma(f)$ of a smooth function can be written as:
$\hat{g} = \sum_{i=1}^n (d x^i)^2 + d f^2$; that is, $\hat g_{ij} = \delta_{ij} +
\sum_{k=1}^{m-n} \frac{\partial f^k}{\partial x^i} \frac{\partial f^k}{\partial x^j}$;
in matrix form, $\hat{g} = I + J^T J$, where Jacobian $J = \nabla f$.
As a submanifold of a Euclidean space with a global parameterization,
the **helicoid** (螺旋曲面) defined by the range of the proper smooth embedding
$F: \mathbb{R}^2 \mapsto \mathbb{R}^3$, $F(u, v) = (u \cos v, u \sin v, v)$,
the pullback metric on the parameter space is $\hat g = F^∗ \bar g = du^2 + (u^2 + 1) dv^2$.

**Product metric** $\oplus_{i=1}^k g_i$ on the product manifold of Riemannian manifolds
is the direct sum of the Riemannian metrics:
$M = \prod_{i=1}^k M_i$, $\forall p \in M$, $\forall v, w \in T_p M$,
$(\oplus_{i=1}^k g_i) |_p (v, w) = \sum_{i=1}^k g_i|_{p_i} (v_i, w_i)$.
Given a smooth chart for each component Riemannian manifold,
the local coordinate representation of the product metric in the direct sum of the charts
equals the matrix direct sum of the local coordinate representations
of the component Riemannian metrics in each chart, i.e. a block diagonal matrix:
$(g) = \text{diag}(g_i)_{i=1}^k$.
**Warped product metric** $g_1 \oplus f^2 g_2$ on the product manifold of two Riemannian manifolds,
given a smooth positive function on the first manifold, is the Riemannian metric defined by
$(g_1 \oplus f^2 g_2) |_p (v, w) = g_1|_{p_1} (v_1, w_1) + f^2(p_1) g_2|_{p_2} (v_2, w_2)$.
Warped product metric is generally not a product metric unless the function is constant.
**Warped product** $M_1 \times_f M_2$ of two Riemannian manifolds,
given a smooth positive function on the first manifold,
is the product manifold endowed with the warped product metric:
$M_1 \times_f M_2 = (M_1 \times M_2, g_1 \oplus f^2 g_2)$.

**Horizontal tangent space** $H_x$ (parallel to base) and
**vertical tangent space** $V_x$ (parallel to fiber, as if fibers are vertical)
at a point in a Riemannian manifold, given a smooth submersion to a smooth manifold,
is the normal space and the tangent space to the fiber containing the point:
$\pi: \tilde M \mapsto M$, let $F = \pi^{-1}(\pi(x))$, then $V_x = T_x F$ and $H_x = (V_x)^\perp$.
Horizontal tangent space has the same dimension as the codomain.
Every tangent space in the domain is the direct sum of
the horizontal and vertical tangent spaces at that point:
$T_x \tilde M = H_x \oplus V_x$.
**Horizontal vector field** $W^H$ / **vertical vector field** $W^V$ on a Riemannian manifold,
given a smooth submersion to a smooth manifold,
is one whose value at each point is a horizontal / vertical tangent vector:
$\forall x \in \tilde{M}$, $W^H_x \in H_x$, $W^V_x \in V_x$.
Every smooth vector field on a Riemannian manifold,
given a smooth submersion to a smooth manifold,
can be written uniquely as the sum of a vertical and a horizontal smooth vector fields:
given $\pi: \tilde M \mapsto M$,
$\mathfrak{X}(\tilde{M}) = \mathfrak{X}^H(\tilde{M}) \oplus \mathfrak{X}^V(\tilde{M})$;
in other words, $\forall W \in \mathfrak{X}(\tilde{M})$,
$\exists! (W^H, W^V) \in \mathfrak{X}^H(\tilde{M}) \times \mathfrak{X}^V(\tilde{M})$: $W = W^V + W^H$.
**Horizontal lift** $\tilde{X}$ of a vector field on the codomain of a smooth submersion π
from a Riemannian manifold is a horizontal vector field on the domain that is π-related to it.
Every smooth vector field on the codomain of a smooth submersion from a Riemannian manifold
has a unique smooth horizontal lift to the domain:
$\forall X \in \mathfrak{X}(M)$, $\exists! \tilde X \in \mathfrak{X}(\tilde M)$:
$d \pi_x (\tilde X_x) = X_{\pi(x)}$.
Every horizontal tangent vector is in the horizontal lift of some vector field:
$\forall x \in \tilde{M}$, $\forall v \in H_x$, $\exists X \in \mathfrak{X}(M)$: $\tilde X_x = v$.
For every tangent vector of the codomain of a smooth submersion from a Riemannian manifold,
there is a unique horizontal tangent vector at each point in the fiber,
that is in the horizontal lift of every smooth extension of the vector:
$\forall (x, v) \in T M$,
define $\tilde v \in \Gamma(\sqcup_{\tilde x \in \pi^{-1}(x)} H_{\tilde x})$:
$\forall \tilde x \in \pi^{-1}(x)$,
$\tilde v_{\tilde x} = (d \pi_{\tilde x}|_{H_{\tilde x}})^{-1}(v)$.
Note that $\forall \tilde x \in \tilde M$,
$d \pi_{\tilde x}|_{H_{\tilde x}} : H_{\tilde x} \mapsto T_{\pi(\tilde x)} M$ is a bijection.
We may call the horizontal vector field $\tilde{v}$ along fiber
the **horizontal lift** of a tangent vector.
For clarity, we may also define a **horizontal lift map** $\tilde{\pi}$
of tangent vectors by a smooth submersion from a Riemannian manifold:
$\tilde{\pi}: \sqcup_{x \in M} (T_x M \times \pi^{-1}(x)) \mapsto
\sqcup_{\tilde x \in \tilde M} H_{\tilde x}$,
$\tilde{\pi}(v, \tilde x) = \tilde v_{\tilde x} = (d \pi_{\tilde x}|_{H_{\tilde x}})^{-1}(v)$.
The horizontal lift map is a bijection.
The notation $\tilde{\pi}$ is inspired by
global pullback $F^∗$ of tangent covectors by a diffeomorphism.

**Riemannian submersion** between Riemannian manifolds
is a smooth submersion whose differential at each point restricts to
a linear isometry from the horizontal tangent space onto the tangent space:
$\pi: \tilde M \mapsto M$, $\forall x \in \tilde{M}$, $d \pi_x|_{H_x}: H_x \cong T_{\pi(x)} M$;
in other words, $\forall v, w \in H_x$, $\tilde g_x(v,w) = g_{\pi(x)}(d \pi_x(v), d \pi_x(w))$.
**Isometric group action** or **group action by isometries** on a Riemannian manifold
is one consisting of isometries on the manifold: $\forall \phi \in G$, $\phi \in \text{Iso}(M, g)$.
An isometric group action on a Riemannian manifold
is identified with a subgroup of the isometries on the manifold,
if different elements of the group define different isometries on the manifold:
$G \subset \text{Iso}(M, g)$.
**Invariant Riemannian metric** under a group acting on a smooth manifold
is one such that the group action is isometric.
**Vertical group action** on a smooth manifold,
given a smooth submersion to another smooth manifold,
is one that takes each fiber to itself: $\forall \phi \in G$, $\pi \circ \phi = \pi$.
**Transitive group action on fibers** on a smooth manifold,
given a smooth submersion to another smooth manifold,
is one that takes each point to a saturated set:
$\forall p \in \tilde M$, $\pi^{-1}(\pi(p)) \subset \{\phi \cdot p: \phi \in G\}$.
*Riemannian submersion theorem*:
Given a surjective smooth submersion from a Riemannian manifold onto a smooth manifold,
if a group acting on the domain is isometric, vertical, and transitive on fibers,
then there is a unique Riemannian metric on the codomain
that makes the submersion a Riemannian submersion.

**Riemannian covering** $\pi: \tilde M \mapsto M$ on a Riemannian manifold
is a smooth covering map that is also a local isometry.
If all the covering automorphisms of a smooth normal covering map on a Riemannian manifold
are isometries, then there is a unique Riemannian metric on the codomain
that makes the covering a Riemannian covering.

Model Riemannian Manifolds have a high degree of symmetry. The most symmetric ones are Euclidean spaces, spheres, and hyperbolic spaces.

Homogeneous ⊃ Isotropic ⊃ Frame-homogeneous

Connected homogeneous ⊃ Symmetric ⊃ Connected frame-homogeneous ⊃ {Euclidean spaces, round spheres, hyperbolic spaces}

**Homogeneous Riemannian manifold** is one whose isometry group acts transitively on the manifold
(cf. homogeneous space):
$\forall p, q \in M$, $\exists \phi \in \text{Iso}(M, g)$: $\phi(p) = q$.
A homogeneous Riemannian manifold looks geometrically the same at every point.
**Isotropy subgroup** $\text{Iso}_p(M, g)$ at a point in a Riemannian manifold
is the subgroup of the isometry group on the manifold where the point is a fixed point.
**Isotropy representation** $I_p: \text{Iso}_p(M, g) \mapsto \text{GL}(T_p M)$
of the isotropy subgroup at a point in a Riemannian manifold
is the map that takes each isometry that is fixed at the point to its differential at the point:
$I_p (\phi) = d \phi_p$.
A Riemannian manifold is **isotropic** at a point
if the isotropy representation of the isotropy subgroup at the point
acts transitively on the set of unit tangent vectors at the point:
$\forall v, w \in T_p M, |v| = |w|$, $\exists \phi \in \text{Iso}_p(M, g)$: $d \phi_p (v) = w$.
**Isotropic Riemannian manifold** is a Riemannian manifold that is isotropic everywhere.
An isotropic Riemannian manifold looks the same in every direction.
Every isotropic Riemannian manifold is homogeneous.
A homogeneous Riemannian manifold is isotropic if it is isotropic at some point.
**Frame-homogeneous Riemannian manifold** is a Riemannian manifold
such that the pushforward of an orthonormal basis by an isometry
is transitive on the set of all orthonormal bases for the tangent spaces on a manifold:
$\forall (e_i), (f_i) \in O(M)$, $\pi(e_i) = p, \pi(f_i) = q$,
$\exists \phi \in \text{Iso}(M, g)$: $\phi(p) = q, d \phi_p (e_i) = f_i$.
Every frame-homogeneous Riemannian manifold is isotropic.
Every frame-homogeneous Riemannian manifold has constant sectional curvature.

**Point reflection** (点反演) at a point in a Riemannian manifold
is an isometry that fixes the point and its differential at the point
flips the sign of each tangent vector:
$\phi \in \text{Iso}_p(M, g)$, $d \phi_p = - \text{Id}$.
**Riemannian symmetric space** is a connected Riemannian manifold
such that there is a point reflection at each point.
Ever Riemannian symmetric space is homogeneous.
A connected homogeneous Riemannian manifold is symmetric if it has a point reflection at some point.
Every connected frame-homogeneous Riemannian manifold is symmetric.
Examples of Riemannian symmetric spaces:
spheres, hyperbolic spaces,
projective spaces, complex projective spaces with the Fubini-Study metric,
Grassmann manifolds.
The irreducible compact Riemannian symmetric spaces are, up to finite covers,
either a compact simple Lie group (SO(2n+1), PSO(2n), PSU(n+1), PSp(n)),
a Grassmannian, a Lagrangian Grassmannian, or a double Lagrangian Grassmannian of subspaces of
$(A \otimes B)^n$, for normed division algebras A and B. [@Huang and Leung, 2011]
A symmetric space is **irreducible** if it is not isometric to a product of symmetric spaces.
**Riemannian locally symmetric space** is a Riemannian manifold
such that there is a point reflection on a neighborhood of each point.

**Euclidean group** $E(n)$ is the semidirect product group
of the Euclidean space (as a Lie group under addition) and the orthogonal group
determined by matrix multiplication: $E(n) = \mathbb{R}^n \rtimes_\theta O(n)$,
where $\theta$ denotes the action of matrix multiplication,
i.e. $\forall x, y \in \mathbb{R}^n$, $\forall P, Q \in O(n)$, $(x, P)(y, Q) = (x + P y, P Q)$.
The n-th Euclidean group has a faithful representation
$\rho: E(n) \mapsto \text{GL}(n+1, \mathbb{R})$ into the (n+1)-th real general linear group
defined by $\rho(x, P) = \begin{pmatrix}P & x \ 0 & 1 \end{pmatrix}$.
The Euclidean group acts on the Euclidean space by $(y, Q) \cdot x = y + Q x$,
which is a subgroup of the isometry group on the Euclidean space,
and its pointwise differentials are transitive on the set of orthonormal bases
for the tangent spaces on the Euclidean space;
thus, every Euclidean space is frame-homogeneous.
The Euclidean group is the full isometry group of the Euclidean space:
$\text{Iso}(\mathbb{R}^n, \bar g) = E(n)$.

**Flat Riemannian metric** on a smooth n-manifold
is a Riemannian metric that makes the manifold locally isometric to the Euclidean n-space.
A Riemannian manifold is flat if and only if
it admits a smooth atlas where every coordinate frame is orthonormal.
Only flat Riemannian manifolds have smooth orthonormal coordinate frames.
Every Riemannian metric on a smooth 1-manifold is flat.
The n-torus as the Riemannian submanifold of the Euclidean 2n-space is flat.
The smooth covering map $X: \mathbb{R}^n \mapsto \mathbb{T}^n$ onto the n-torus,
defined by $X(u^i)_{i=1}^n = (\cos u^i, \sin u^i)_{i=1}^n$,
is a local isometry and thus a Riemannian covering.
A (pseudo-)Riemannian manifold is flat if and only if its curvature tensor is zero.

**Round metric** $\mathring g_R$ on the sphere $\mathbb{S}^n(R)$ of radius $R$
centered at the origin in the Euclidean (n+1)-space
is the induced metric on the submanifold:
$\mathring g_R = \iota^∗ \bar g$ where $\iota: \mathbb{S}^n(R) \mapsto \mathbb{R}^{n+1}$.
The (n+1)-th orthogonal group is the full isometry group of an n-sphere of any radius:
$\text{Iso}(\mathbb{S}^n(R), \mathring g_R) =O(n+1)$.
Every round sphere is frame-homogeneous.
Spheres of radius $R$ have constant sectional curvature $R^{-2}$.
**Great circle** on the n-sphere is its intersection with
any two-dimensional linear subspace of the Euclidean (n+1)-space:
$\mathbb{S}^n(R) \cap \Pi$, where $\Pi \in G(2, n+1)$.
A nonconstant curve on a sphere is a maximal geodesic if and only if
it is a periodic constant-speed curve whose image is a great circle.
Every sphere is geodesically complete.

**Conformally related** or **pointwise conformal** Riemannian metrics on a smooth manifold
are ones that differ by a smoothly varying scaling factor:
$\exists f \in C^\infty(M)$, $g_2 = f g_1$.
**Conformal diffeomorphism** between Riemannian manifolds is a diffeomorphism such that
the pullback metric is conformal to the Riemannian metric on the domain:
$\phi: M \mapsto \tilde M$, $\exists f \in C^\infty(M)$, $\phi^∗ \tilde g = f g$.
Conformal (共形) diffeomorphisms are the same as angle-preserving (保角) diffeomorphisms.
**Conformally equivalent** Riemannian manifolds are ones related by a conformal diffeomorphism.
**Locally conformally flat** Riemannian metric on a smooth n-manifold
is a Riemannian metric that makes some open subsets that cover the manifold
conformally equivalent to open subsets of the Euclidean n-space.

**Stereographic projection** (球极平面投影)
$\sigma: \mathbb{S}^n(R) \setminus \{N\} \mapsto \mathbb{R}^n$
from the north pole of the n-sphere of radius $R$ to the Euclidean n-space
is the map that sends each point on the sphere other than the north pole to the point
where the line through the north pole and the point intersects the equatorial hyperplane:
$\sigma(x^i)_{i=1}^{n+1} = \frac{R}{R - x^{n+1}} (x^i)_{i=1}^n$.
The stereographic projection is a conformal diffeomorphism between
the n-sphere of radius $R$ minus the north pole and the Euclidean n-space.
The round metric on a sphere of any radius is not flat, but is locally conformally flat.

**Hyperbolic space** of radius $R$ and dimension $n \ge 2$
is any of the mutually isometric Riemannian manifolds defined as follows:
(1) *Hyperboloid model* $(\mathbb{H}^n (R), \breve g_R^1)$:
the submanifold of the (n+1)-dimensional Minkowski space
defined by the upper sheet of the hyperboloid $\sum_{i=1}^n (\xi^i)^2 - \tau^2 = -R^2$
(with the induced metric).
(2) *Beltrami-Klein model* $(\mathbb{K}^n (R), \breve g_R^2)$:
the ball of radius $R$ centered at the origin of the Euclidean n-space, with metric
$\breve g_R^2 = \frac{R^2}{R^2 - |w|^2} \left[\sum_{i=1}^n (d w^i)^2 +
\frac{(\sum_{i=1}^n w^i d w^i)^2}{R^2 - |w|^2} \right]$.
(3) *Poincare ball model* $(\mathbb{B}^n (R), \breve g_R^3)$:
the ball of radius $R$ centered at the origin of the Euclidean n-space, with metric
$\breve g_R^3 = \frac{4 R^4}{(R^2 - |u|^2)^2} \sum_{i=1}^n (d u^i)^2$.
(4) *Poincare half-space model* $(\mathbb{U}^n (R), \breve g_R^4)$:
the upper half-space in the Euclidean n-space, with metric
$\breve g_R^4 = \frac{R^2}{(x^n)^2} \sum_{i=1}^n (d x^i)^2$.
**Hyperbolic plane** is a 2-dimensional hyperbolic space.
Every Hyperbolic space is frame-homogeneous.
Hyperbolic spaces of radius $R$ have constant sectional curvature $-R^{-2}$.

**Lorentz group** $O(n, 1)$ of dimension $n+1$...
**Orthochronous Lorentz group** $O^+(n, 1)$ of dimension $n+1$...
The (n+1)-dimensional orthochronous Lorentz group
is the full isometry group of an n-dimensional Hyperbolic space of any radius:
$\text{Iso}(\mathbb{H}^n(R), \breve g_R) = O^+(n, 1)$.

**Central projection** from the hyperboloid to the ball...
**Hyperbolic stereographic projection**...
**Cayley transform**...
**Generalized Cayley transform**...
Every hyperbolic space is locally conformally flat.

**Great hyperbola** in the hyperboloid model of a hyperbolic space
is the intersection of the hyperboloid with a 2-dimensional linear subspace.
A nonconstant curve in a hyperbolic space is a maximal geodesic if and only if
it is a constant-speed embedding of the real line whose image is one of the following:
(1) Hyperboloid model: a great hyperbola;
(2) Beltrami-Klein model: the interior of a line segment with endpoints on the sphere;
(3) Poincare ball model: the interior of a diameter,
or the arc of a Euclidean circle that intersects the sphere orthogonally;
(4) Half-space model: a half-line orthogonal to the boundary,
or a Euclidean half-circle with center on the boundary.
Every hyperbolic space is geodesically complete.

Riemannian submanifolds of Euclidean spaces are particularly interesting. We already know that: the Euclidean space $(\mathbb{R}^n, \bar{g}, \bar{\nabla}, \overline{Rm})$ has Euclidean metric $\bar g_{ij} = \delta_{ij}$, Euclidean connection $\bar \nabla_X Y = (\nabla Y) X$ where $\nabla Y$ is the Jacobian, and curvature tensor $\overline{Rm} = 0$; its submanifold $(M, g, \nabla^\top, Rm)$ has induced metric $g_p = \bar g_p|_{(T_p M)^2}$, tangential connection $\nabla^\top_X Y = \pi^\top (\bar \nabla_{\tilde X} \tilde Y)$; for a graph parameterization $\Phi(x) = (x, f(x))$ of the submanifold, the induced Riemannian manifold on the parameter space is $(X, \hat g, \hat \nabla)$, with metric $\hat{g} = I + J^T J$, where $J = \nabla f$ is the Jacobian. Covariant derivative of a vector field along a curve $D_t V = \pi^\top(V')$; in particular, acceleration of a curve equals the tangential acceleration: $D_t \gamma' = \pi^\top(\gamma'')$. In this section we will show that the submanifold has curvature tensor $Rm(W,X,Y,Z) = (\text{II}(W, Z), \text{II}(X, Y)) - (\text{II}(W, Y), \text{II}(X, Z))$, where second fundamental form $\text{II}(X, Y) = \pi^\perp (\bar\nabla_{\tilde X} \tilde Y)$. Note that $Rm_p(v,w,w,v) = (\text{II}(v, v), \text{II}(w, w)) - \|\text{II}(v, w)\|^2$ and $\text{II}(v, v) = \gamma_v''(0)$, we can compute the sectional curvature by: $\sec(v, w) = [(\gamma_v''(0), \gamma_w''(0)) - \|\text{II}(v, w)\|^2] / [\|v\|^2 \|w\|^2 - (v, w)^2]$, where $\text{II}(v, w) = \pi^\perp (\bar\nabla_{\tilde v} \tilde w)$.

**Normal vector** at a point on a smooth submanifold of a Riemannian manifold
is a tangent vector of the manifold that is normal to the tangent space of the submanifold:
$v \in T_p \tilde{M}$, $\forall w \in T_p M$, $\langle v, w \rangle_g = 0$.
**Normal vector field** along a smooth submanifold of a Riemannian manifold
is a section of the ambient tangent bundle $T \tilde{M}|_M$
whose value at each point is a normal vector to the submanifold.
**Normal space** $N_p M$ at a point on a smooth submanifold of a Riemannian manifold
is the vector space consisting of all the tangent vectors normal to the submanifold at the point.
The tangent space of a Riemannian manifold at each point of a smooth submanifold
is the orthogonal direct sum of the normal space and the tangent space of the submanifold
at that point: $N_p M = (T_p M)^\perp$.
**Normal bundle** $N M$ of a smooth submanifold of a Riemannian manifold
is the disjoint union of all its normal spaces: $N M = \sqcup_{p \in M} N_p M$.
The normal bundle of a smooth n-submanifold of a Riemannian m-manifold
is a smooth rank-$(m-n)$ vector subbundle of the ambient tangent bundle.
The normal bundle $N \partial M$ to the boundary of a smooth Riemannian manifold
is a smooth rank-1 vector bundle,
and there is a unique smooth outward-pointing unit normal vector field along the boundary.
**Tangential projection** $\pi^\top: T \tilde{M}|_M \mapsto T M$
is the smooth bundle homomorphism whose restriction to each fiber
is the orthogonal projection from the ambient tangent space to the tangent space of the submanifold.
**Normal projection** $\pi^\perp: T \tilde{M}|_M \mapsto N M$
is the smooth bundle homomorphism whose restriction to each fiber
is the orthogonal projection from the ambient tangent space to the normal space of the submanifold.
We often use shorthand notations $\pi^\top(X) = X^\top$ and $\pi^\perp(X) = X^\perp$,
where $X \in \Gamma(T \tilde M |_M)$.

**Second fundamental form** $\text{II}$ (as in bilinear form; read "two")
of an embedded Riemannian submanifold of a (pseudo-)Riemannian manifold
is a smooth field of symmetric bilinear operators identified with the symmetric bilinear map
$\text{II}: \mathfrak{X}^2(M) \mapsto \Gamma(N M)$, $\forall X, Y \in \mathfrak{X}(M)$,
$\text{II}(X, Y) = \pi^\perp (\tilde\nabla_{\tilde X} \tilde Y)$,
where $\tilde X|_M = X$ and $\tilde Y|_M = Y$ are arbitrary extensions.
Let $E$ be the smooth vector bundle over the submanifold with fibers
$E_p = \mathcal{L}((T_p M)^2, N_p M)$,
then the second fundamental form is a smooth section of the bundle: $\text{II} \in \Gamma(E)$;
at every point on the submanifold, it is a symmetric bilinear operator
from the Cartesian square of the tangent space to the normal space:
$\text{II}_p \in \mathcal{L}((T_p M)^2, N_p M)$.
The second fundamental form is completely determined
by its action on pairs of the same unit tangent vectors.
The second fundamental form provides the difference between
the intrinsic and the ambient Levi-Civita connections:
(1) *Gauss Formula*:
$\forall X, Y \in \mathfrak{X}(M)$, $\tilde\nabla_X Y = \nabla_X Y + \text{II}(X, Y)$;
(2) *Gauss Formula along a Curve*:
$\forall \gamma \in C^\infty(I, M)$, $\forall X \in \mathfrak{X}(\gamma)$,
$\tilde D_t X = D_t X + \text{II}(\gamma', X)$.
*Gauss Equation*:
The second fundamental form provides the difference between
the intrinsic and the ambient curvature tensors:
$\forall W, X, Y, Z \in \mathfrak{X}(M)$, $\widetilde{Rm}(W,X,Y,Z) - Rm(W,X,Y,Z) =$
$\langle \text{II}(W, Y), \text{II}(X, Z) \rangle - \langle \text{II}(W, Z), \text{II}(X, Y) \rangle$.

Geodesic **curvature** $\kappa$ of a smooth unit-speed curve in a (pseudo-)Riemannian manifold
is the non-negative function that gives the norms of the acceleration vectors:
$\kappa(t) = |D_t \gamma'(t)|$.
Geodesic **curvature** of a regular curve in a Riemannian manifold
is the curvature of a unit-speed reparameterization of the curve at the corresponding point:
$\kappa(t) = \bar{\kappa} \circ \phi^{-1}(t)$, where $\bar{\gamma} = \gamma \circ \phi$.
Geodesic **curvature** of a nonzero-speed regular curve in a pseudo-Riemannian manifold
is defined similarly.
Geodesic has zero geodesic curvature.
For a regular curve in an embedded Riemannian submanifold of a (pseudo-)Riemannian manifold,
its **intrinsic curvature** $\kappa$ is its geodesic curvature as a curve in the submanifold,
and its **extrinsic curvature** $\tilde \kappa$ is its geodesic curvature
as a curve in the ambient manifold:
for unit-speed curves, $\tilde\kappa(t) = |\tilde D_t \gamma'(t)|$;
reparameterization should be applied otherwise.
*Geometric interpretation of the second fundamental form*:
(1) The action of the second fundamental form on any tangent vector of the submanifold
is the initial ambient acceleration of the geodesic with that initial velocity:
$\forall v \in T_p M$, $\tilde D_t \gamma_v'(0) = \text{II}(v, v)$;
(2) For a unit tangent vector, the norm of the second fundamental form
is the initial ambient/extrinsic curvature of the geodesic:
if $|v| = 1$, then $\tilde\kappa(0) = |\text{II}(v, v)|$.
**Totally geodesic** submanifold of a Riemannian manifold is one such that
every ambient geodesic that is tangent to the submanifold at some time
stays in the submanifold for an open interval containing that time.
For an embedded Riemannian submanifold of a (pseudo-)Riemannian manifold,
the following are equivalent:
(a) it is totally geodesic;
(b) every geodesic is an ambient geodesic;
(c) its second fundamental form vanishes identically, i.e. it is identically zero.

**Second fundamental form** $\text{II}_N$ in the direction of a normal vector field
is the symmetric 2-tensor field that provides the inner product of
the second fundamental form and the normal vector field:
$N \in \Gamma(N M)$, $\forall X, Y \in \mathfrak{X}(M)$,
$\text{II}_N(X, Y) = \langle N, \text{II}(X, Y) \rangle$.
**Weingarten map** $W_N$ in the direction of a normal vector field
is the self-adjoint, smooth bundle homomorphism on the tangent bundle such that:
$\langle W_N(X), Y \rangle = \langle N, \text{II}(X, Y) \rangle$;
note that $\langle W_N(X), Y \rangle = \text{II}_N(X, Y)$.
*Weingarten Equation*:
The Weingarten map of a vector field in the direction of a normal vector field
equals the opposite of the tangential component of the ambient covariant derivative of
the normal vector field in the direction of the vector field:
$\forall X \in \mathfrak{X}(M)$, $\forall N \in \Gamma(N M)$,
$\pi^\top(\tilde \nabla_X N) = - W_N(X)$.
**Normal connection** $\nabla^\perp$
of an embedded Riemannian submanifold of a (pseudo-)Riemannian manifold
is the connection on its normal bundle defined by:
$\forall X \in \mathfrak{X}(M)$, $\forall N \in \Gamma(N M)$,
$\nabla^\perp_X N = \pi^\perp (\tilde\nabla_X \tilde N)$, where $\tilde N$ is a smooth extension.
The normal connection is compatible with the ambient (pseudo-)Riemannian metric:
$X \langle N_1, N_2 \rangle = \langle \nabla^\perp_X N_1, N_2 \rangle +
\langle N_1, \nabla^\perp_X N_2 \rangle$.
Define a connection $\nabla^E$ in the bundle $E$ over the submanifold:
$\forall X, Y, Z \in \mathfrak{X}(M)$, $\forall B \in \Gamma(E)$,
$(\nabla^E_X B)(Y, Z) = \nabla^\perp_X (B(Y, Z)) - B(\nabla_X Y, Z) - B(Y, \nabla_X Z)$.
*Codazzi Equation* [@Peterson1853; @Mainardi1856; @Codazzi1868]:
The normal part of an ambient curvature endomorphism
equals a difference between covariant derivatives of the second fundamental form:
$(\tilde R(X, Y) Z)^\perp = (\nabla^E_X \text{II})(Y, Z) - (\nabla^E_Y \text{II})(X, Z)$.

Now we consider a hypersurface of a Riemannian (n+1)-manifold, and all constructs depend on
the choice of a smooth unit normal vector field $N$ on the hypersurface unless otherwise specified.
**Scalar second fundamental form** $h$ of the hypersurface
is its second fundamental form in the direction of that normal vector field:
$h = \text{II}_N$; $\text{II}(X, Y) = h(X, Y) N$;
or equivalently, $h(X, Y) = \langle N, \tilde \nabla_X Y \rangle$.
**Shape operator** $s$ of the hypersurface is the (1,1)-tensor field
obtained from the scalar second fundamental form by raising an index:
$s = h^\sharp$; i.e. its Weingarten map: $s = W_N$; or equivalently,
a self-adjoint endomorphism on the tangent bundle such that
$\forall X, Y \in \mathfrak{X}(M)$, $\langle s X, Y \rangle = h(X, Y)$.
**Principal curvatures** $(\kappa_i)_{i=1}^n$ at a point of a hypersurface
are the eigenvalues of the shape operator.
Principal curvatures are determined up to sign, depending on the direction of normal vector.
**Principal directions** at a point of a hypersurface w.r.t. a principal curvature
is the corresponding eigenspace.
For every point on the hypersurface,
there is an isometry from a neighborhood of the hypersurface to the graph of a function
such that the coefficients of the quadratic terms are the principal curvatures at that point:
$f(x) = \frac{1}{2} \sum_{i=1}^n \kappa_i x_i^2 + \mathcal{O}(|x|^3)$.
**Gaussian curvature** $K$ at a point of a hypersurface
is the product of the principal curvatures: $K = \det(s)$; that is, $K = \prod_{i=1}^n \kappa_i$,
its sign is independent of the normal vector field if $n$ is an even number.
For a surface, its Gaussian curvature is the product of its two principal curvatures,
whose sign is easy to check for quadric surfaces (aka quadrics):
(1) zero: cylinder;
(2) positive: ellipsoid, elliptic paraboloid, elliptic hyperboloid of two sheets;
(3) negative: hyperbolic paraboloid, elliptic hyperboloid of one sheet.
*Gauss's Theorema Egregium* (绝妙定理):
The Gaussian curvature at every point of an embedded Riemannian 2-submanifold of the Euclidean 3-space
is equal to one-half the scalar curvature of the induced metric at that point,
and thus the Gaussian curvature is a local isometry invariant of the submanifold:
$K(p) = S(p) / 2$.
**Mean curvature** $H$ at a point of a hypersurface
is the average of the principal curvatures: $H = \text{tr}(s) / n$;
that is, $H = \sum_{i=1}^n \kappa_i / n$.
Unlike the Gaussian curvature, the mean curvature is not a local isometry invariant:
a cylinder is locally isometric to the Euclidean plane, but their mean curvatures are different.
**Kulkarni–Nomizu product** $h \mathbin{\bigcirc\mspace{-15mu}\wedge} k$ of symmetric 2-tensors
is the 4-tensor defined by:
$h \mathbin{\bigcirc\mspace{-15mu}\wedge} k (w, x, y, z) = h(w, z) k(x, y) + h(x, y) k(w, z)$
$- h(w, y) k(x, z) - h(x, z) k(w, y)$.
**Exterior covariant derivative** $D T$ of a smooth symmetric 2-tensor field
is the 3-tensor field defined by: $(D T)(x, y, z) = -(\nabla T)(x, y, z) + (\nabla T)(x, z, y)$.
**Codazzi tensor** is a symmetric 2-tensor field whose exterior covariant derivative is zero.
*Fundamental Equations for a Hypersurface*:
(1) Gauss formula: $\tilde\nabla_X Y = \nabla_X Y + h(X, Y) N$;
(2) Gauss formula for a curve: $\tilde D_t X = D_t X + h(\gamma', X) N$;
(3) Gauss equation: $\widetilde{Rm} - Rm = - h \mathbin{\bigcirc\mspace{-15mu}\wedge} h / 2$;
(4) Weingarten equation: $\tilde \nabla_X N = - s X$;
(5) Codazzi equation: $\widetilde{Rm}(X, Y, Z, N) = (D h)(Z, X, Y)$.
*Fundamental Equations for a Hypersurface of a Euclidean Space*:
(3) Gauss equation: $Rm = h \mathbin{\bigcirc\mspace{-15mu}\wedge} h / 2$;
(5) Codazzi equation: $D h = 0$.
This means, for hypersurfaces of Euclidean Spaces,
its scalar second fundamental form completely determines its curvature tensor,
and its scalar second fundamental form is a Codazzi tensor.
In a local frame, these equations read:
(3) Gauss equation: $R_{ijkl} = h_{il} h_{jk} - h_{ik} h_{jl}$;
(5) Codazzi equation: $h_{ij;k} = h_{ik;j}$,
where the semicolon separates the index of differentiation.
With a smooth local parameterization $X \in C^\infty(U, \mathbb{R}^{n+1})$,
the scalar second fundamental form can be computed as: $\forall i, j \in n$,
$h(\partial_i X, \partial_j X) = \langle \frac{\partial^2 X}{\partial u^i \partial u^j}, N \rangle$.
If a smooth normal vector field can be obtained, the shape operator can be computed as:
$\forall X \in \mathfrak{X}(M)$,
$s X = - \bar\nabla_X N = - \sum_{i,j=1}^{n+1} X^j (\partial_j N^i) \partial_i$.

**Gaussian curvature** $K$ of a Riemannian 2-manifold is half the scalar curvature: $K := S / 2$.
**k-plane** $\Pi$ in a vector space is a k-dimensional linear subspace:
$\Pi \in G_k(V)$, where $G$ refers to Grassmann manifold.
A k-plane may also be specified by a basis of the subspace, e.g.
**(v,w)-plane** is the 2-plane spanned by linearly independnet vectors $v$ and $w$ in a linear space.
**Plane section** $S_\Pi$ determined by a 2-plane in a tangent plane of a Riemannian n-manifold,
$n \ge 2$, is the embedded 2-dimensional submanifold defined by
a diffeomorphic image of the 2-plane via the exponential map:
$\Pi \in G_2(T_p M)$, $V \subset T_p M$, $U \subset M$, $\exp_p: V \cong U$,
$S_\Pi = \exp_p(\Pi \cap V)$.
**Sectional curvature** $\text{sec}(\Pi)$ of a 2-plane in a tangent plane of a Riemannian manifold
is the intrinsic Gaussian curvature at the point of the plane section determined by the 2-plane;
"intrinsic" means using the induced metric on the plane section.
**Sectional curvature** $\text{sec}(v, w)$ given two linearly independent tangent vectors at a point
is sectional curvature of the (v,w)-plane: $\text{sec}(v, w) = Rm_p(v, w, w, v) / |v \wedge w|^2$,
where $|v \wedge w| = \sqrt{|v|^2 |w|^2 - \langle v, w \rangle^2}$.
*Sectional curvatures determine the curvature tensor*:
Two algebraic curvature tensors on a finite-dimensional inner product space are identical,
if they provide the same sectional curvature for every pair of linearly independent vectors.
*Geometric interpretation of Ricci and scalar curvatures*:
The action of the Ricci curvature on every unit tangent vector of a Riemannian n-manifold, $n \ge 2$,
is the sum of the sectional curvatures of the 2-planes spanned by this tangent vector
and each of $(n - 1)$ other vectors that together form an orthonormal basis of the tangent space;
and the scalar curvature at every point is the sum of all sectional curvatures of the 2-planes
spanned by ordered pairs of distinct basis vectors in any orthonormal basis:
$\forall p \in M$, $\forall (v_i)_{i=1}^n \in O(T_p M)$,
$Rc_p(v_i, v_i) = \sum_{j \ne i} \sec(v_i, v_j)$,
$S(p) = \sum_{i=1}^n \sum_{j \ne i} \sec(v_i, v_j)$.
If the sectional curvatures of a Riemannian manifold are sign-definite
(i.e. positive, negative, nonnegative, or nonpositive),
then its Ricci and scalar curvatures both have the same sign-definiteness;
note that the Ricci tensor would be called positive-definite, negative-definite,
positive semi-definite, or negative semi-definite.
**Constant sectional curvature** Riemannian metric or Riemannian manifold is one
whose sectional curvatures are the same for all planes at all points.
A Riemannian metric has constant sectional curvature $c$ if and only if
its curvature tensor can be written as $Rm = \frac{c}{2} g \mathbin{\bigcirc\mspace{-15mu}\wedge} g$,
or in any local basis, $R_{ijkl} = c (g_{il} g_{jk} - g_{ik} g_{jl})$.
For a Riemannian n-manifold with constant sectional curvature $c$:
the Ricci tensor $Rc = (n-1) c g$, or in a local basis, $R_{ij} = (n-1) c g_{ij}$;
the scalar curvature $S = n (n-1) c$;
the Riemann curvature endomorphism $R(v, w) x = c (\langle w, x \rangle v - \langle v, x \rangle w)$.

*Nash embedding theorem* ($C^1$ version) [@Nash1954]:
If a Riemannian manifold $(M, g)$ admits a short smooth embedding $f$ into $\mathbb{R}^n$,
then it admits isometric $C^1$ embeddings into $\mathbb{R}^n$ that approximate $f$ uniformly.
As follows from the Whitney embedding theorem,
any Riemannian manifold of dimension $m$ admits an isometric $C^1$ embedding
into an arbitrarily small neighborhood in $\mathbb{R}^{2m}$.

*Myers Theorem* [@Myers1941]:
If a Riemanniann n-manifold is complete and its Ricci curvature has a positive lower bound,
then it is compact and its fundamental group is finite:
$\forall v \in T M$, $|v| = 1$, $Rc(v, v) \ge (n-1) R^{-1}$,
then $\text{diam}(M) \le \pi R$ and $\forall p \in M$, $|\pi_1(M, p)| \in \mathbb{N}$.
Note that noncompact Riemanniann manifolds can be complete and
have strictly positive Ricci or even sectional curvature,
as long as the curvature gets arbitrarily close to zero, e.g. the paraboloid.

Symplectic geometry or symplectic topology is the study of invariant properties of symplectic manifolds under symplectomorphisms.

**Symplectic tensor** (辛张量) $\omega$ (aka "symplectic form") is a nondegenerate 2-covector.
**Symplectic vector space** $(V, \omega)$ is a vector space endowed with a symplectic tensor.
Any symplectic vector space is even-dimensional.
**Symplectic basis** for a symplectic vector space is a basis $(a_i, b_i)_{i=1}^n$,
$n = (\dim V) / 2$, such that its symplectic tensor has the canonical form
$\omega = \sum_{i=1}^n \alpha_i \wedge \beta_i$,
where $(\alpha_i, \beta_i)_{i=1}^n$ is the corresponding basis for its dual space.
Any symplectic vector space has a symplectic basis.

**Symplectic complement** $S^\perp$ of a linear subspace $S$ of a symplectic vector space
is the subspace consisting of all vectors such that
the action of the symplectic tensor on them and those in $S$ are always zero:
$S^\perp = \{v \in V : \forall w \in S, \omega(v, w) = 0\}$.
**Symplectic subspace** of a symplectic vector space is a linear subspace
which intersects with its symplectic complement only at the origin: $S \cap S^\perp = \{0\}$.
**Isotropic subspace** / **Coisotropic subspace** / **Lagrangian subspace**
of a symplectic vector space is a linear subspace
that is included in / includes / equals its symplectic complement, respectively.
The dimension of a symplectic vector space equals the sum of the dimensions of
any linear subspace and its symplectic complement: $\dim V = \dim S + \dim S^\perp$.
Thus, a linear subspace can be isotropic / isotropic / Lagrangian only if
its dimension is no greater than / is no less than / equals half the dimension of the full space.

**Nondegenerate 2-form** on a smooth manifold
is a 2-form whose value is a nondegenerate 2-covector everywhere.
**Symplectic form** (辛形式) or **symplectic structure** on a smooth manifold
is a closed nondegenerate 2-form.
**Symplectic manifold** (辛流形) $(M, \omega)$ is a smooth manifold endowed with a symplectic form.
**Symplectic coordinates**, **canonical coordinates**, or **Darboux coordinates**
for a symplectic 2n-manifold are coordinates $(x_i, y_i)_{i=1}^n$
in which the symplectic form of the manifold has the canonical coordinate representation
$\omega = \sum_{i=1}^n d x^i \wedge d y^i$.
*Darboux theorem*: Every symplectic manifold has a smooth atlas
where its symplectic form has the canonical form in every chart.

**Symplectomorphism** (辛微分同胚) $F: M_1 \mapsto M_2$
is a diffeomorphism between symplectic manifolds
by which the pullback of the symplectic form of its codomain is that of its domain:
$F^∗ \omega_2 = \omega_1$.

Hamiltonian vector fields. Hamiltonian system, a canonical system of ordinary differential equations corresponding to any smooth real-valued function on a symplectic manifold. Poisson brackets.

Hamiltonian flowout theorem.

**Symplectic submanifold** of a symplectic manifold
is a smooth submanifold whose tangent spaces are symplectic subspaces.
**Isotropic submanifold**, **coisotropic submanifold**, and **Lagrangian submanifold**
are analogously defined.

Constraint as a submanifold (geometry; constraint manifold) or a subbundle (force).
**Holonomic constraint** (完整约束) on the tangent bundle is an integrable subbundle.

Contact structure is an odd-dimensional analogue of symplectic structures. Many constructs for symplectic manifolds have analogues in contact geometry.

**Contact form** (切触形式) $\theta$ on a smooth (2n+1)-manifold $M$
is a nonvanishing smooth 1-form such that
the restriction of its differential $d \theta_p \in T^2(T_p^∗ M)$ at each point
to its kernel $\text{Ker}(\theta_p)$ is nondegenerate, i.e. a symplectic tensor.
**Contact coordinates** for a contact form on a smooth (2n+1)-manifold
are smooth coordinates $(x_i, y_i, z)_{i=1}^n$ in which
the contact form has the canonical coordinate representation
$\theta = d z - \sum_{i=1}^n y^i d x^i$.
*Contact Darboux theorem*: Every contact form on a smooth (2n+1)-manifold
can be put in the canonical form in charts that form a smooth atlas.

**Contact structure** (接触结构) $H$ on a smooth (2n+1)-manifold
is a smooth rank-$2n$ distribution whose smooth local defining forms are all contact forms.
Contact form for a contact structure is a local defining form for the contact structure.
**Contact manifold** (切触流形) $(M, H)$ is a smooth manifold endowed with a contact structure.
Every oriented compact smooth 3-manifold admits a contact structure [@Martinet1971].

Contact Hamiltonian vector field. Contact flowout theorem.

**Isotropic submanifold** of a contact manifold is a smooth submanifold
whose tangent bundle is contained in the contact structure of the manifold.
**Legendrian submanifold** of a contact (2n+1)-manifold is an isotropic n-submanifold.
**Legendrian section** of the 1-jet bundle of a smooth manifold
is a smooth section $\eta: M \mapsto J^1 M$ whose range is a Legendrian submanifold.
A smooth local section of the 1-jet bundle of a smooth manifold is the 1-jet of a smooth function
if and only if it is a Legendrian section.