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 $q$ on a finite-dimensional vector space is a nondegenerate bilinear form. 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 of a scalar product space is defined in the same way as in an inner product space. 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. A tuple of vectors in a scalar product space is nondegenerate if its 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 $s$ 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. 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: pseudo-Euclidean space, pseudosphere, pseudohyperbolic space. 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. pseudosphere... pseudohyperbolic 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.
Connection... Levi-Civita connection...
Exponential map $\exp: \mathscr{E} \mapsto M$ is defined by $\exp{v} = \gamma_v(1)$, where $\mathscr{E}$ is the subset of the tangent bundle $T M$ such that $\gamma_v(1)$ is defined on an interval containing $[0,1]$. Restricted exponential map $\exp_p$ at a point $p \in M$ is the restriction of the exponential map to the tangent space at $p$: $\exp_p = \exp|_{\mathscr{E} \cap T_pM}$.
Normal neighborhoods... (Riemannian) normal coordinates. 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.
Curvature is the fundamental invariant of a pesudo-Riemannian metric.
Riemann curvature endomorphism or (1,3)-curvature tensor $R: \mathfrak{X}^3(M) \mapsto \mathfrak{X}(M)$ on a pseudo-Riemannian manifold... Riemann curvature tensor $Rm$ or $Riem$ on a pseudo-Riemannian manifold is the (0,4)-tensor field obtained from the curvature endomorphism by lowering an index: $Rm = R^\flat$. The curvature tensor of a pseudo-Riemannian manifold is a local isometry invariant.
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. 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.
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: $g_{ij} = \delta_{ij}$.
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: $V \cong \mathbb{R}^n$. 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: $\bar g_{ij} = \delta_{ij}$.
Riemannian metric determines the isometric invariants of a Riemannian manifold. Different metrics on the same manifold can have vastly different geometric properties. [@Lee2012 Problem 13-20]
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. Isometric immersion / isometric embedding / local isometry between Riemannian manifolds is a smooth immersion / smooth embedding / local diffeomorphism analogously defined. 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.
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 defined by $\theta = \arccos \left(\frac{\langle v, w \rangle_g}{|v|_g |w|_g}\right)$. Norm and angle (as smooth tensor fields) on a Riemannian manifold are local isometry invariants.
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 parametrization of the curve. Length of piecewise regular curve segments in a Riemannian manifold is a local isometry invariant. 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: $s(t) = \int_a^t |\gamma'(t)|_g dt$. 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 parametrized if it is unit-speed. A piecewise regular curve segement is parametrized by arc length if it is unit-speed and its domain starts from zero. Every regular curve in a Riemannian manifold has a unit-speed forward reparametrization. Every piecewise regular curve segement in a Riemannian manifold has a unique forward reparametrization by arc length.
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: $\sup_{x,y \in M} d_g(x, y) < \infty$. Complete Riemannian manifold is a connected Riemannian manifold if the metric space it generates is complete, i.e. every Cauchy sequence converges. Complete Riemannian metric is the Riemannian metric of a complete 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.
Geodesic...
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 $p \in (0, 1)$. (How to estimate the Riemmannian metric compatible with a distance function?)
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 $\iota^∗ \tilde{g}$, or first fundamental form (a symmetric 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: $g = \iota^∗ \tilde{g}$.
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) at a point in a Riemannian manifold, given a smooth submersion $\pi$ to a smooth manifold, is the normal space and the tangent space to the fiber containing the point: $H_x = (V_x)^\perp$, $V_x = T_x \pi^{-1}(\pi(x))$. Horizontal tangent space has the same dimension as the codomain. 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: $\forall x \in \tilde{M}$, $\forall x \in \tilde{M}, d \pi_x: H_x \cong T_{\pi(x)} M$; i.e. $\forall v, w \in H_x$, $\tilde g_x(v,w) = g_{\pi(x)}(d \pi_x(v), d \pi_x(w))$. 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: $\forall p \in \tilde M$, $\pi^{-1}(\pi(p)) \subset {\phi \cdot p: \phi \in G}$. If a group acting on a Riemannian manifold is isometric, vertical, and transitive on fibers, given a surjective smooth submersion onto a smooth manifold, 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: Euclidean spaces, spheres, and hyperbolic spaces.
Homogeneous Riemannian manifold is one whose isometry group acts transitively on the manifold: $\forall p, q \in M$, $\exists \phi \in \text{Iso}(M, g)$: $\phi(p) = q$. 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 ponit 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: $\forall (e_i), (f_i) \in O(M)$, $\pi(e_i) = p, \pi(f_i) = q$, $\exists \phi \in \text{Iso}(M, g)$: $\phi(p) = q, d \phi_p (e_i) = f_i$. Any frame-homogeneous Riemannian manifold is isotropic.
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. 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. Flat Riemannian manifolds have zero curvature.
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.
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.
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: $\text{Iso}(\mathbb{H}^n(R), \breve g_R) = O^+(n, 1)$. Every round sphere is frame-homogeneous.
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.
The metric induced by a smooth local parametrization $X: \phi(U) \mapsto \tilde{M}$ of a Riemannian submanifold $(M, g)$ equals the pullback of the Riemannian metric of the ambient manifold: $X^∗ g = X^∗ \tilde{g}$. For any Riemannian n-submanifold of the Euclidean m-space, the metric induced a smooth local parametrization $X: \phi(U) \mapsto \mathbb{R}^m$ can be written as $X^∗ \bar g = \sum_{i=1}^m (d X^i)^2$ which expands to $\sum_{i,j=1}^n \sum_{k=1}^m \frac{\partial X^k}{\partial x^i} \frac{\partial X^k}{\partial x^j} d x^i d x^j$, i.e. the local coordinate representation of the Riemannian metric is $g_{ij} = \sum_{k=1}^m \frac{\partial X^k}{\partial x^i} \frac{\partial X^k}{\partial x^j}$. In particular, the metric induced the graph parametrization $X(x) = (x, f(x))$ of the graph $\Gamma(f)$ of a smooth function can be written as: $X^∗ \bar g = \sum_{i=1}^n (d x^i)^2 + d f^2$.
Consider the proper smooth embedding $F: \mathbb{R}^2 \mapsto \mathbb{R}^3$, $F(u, v) = (u \cos v, u \sin v, v)$, whose range is known as a helicoid (螺旋曲面), the pullback metric of the Euclidean metric is $F^∗ \bar g = du^2 + (u^2 + 1) dv^2$.
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.
Second fundamental form (a symmetric bilinear form) $\text{II}: \mathfrak{X}^2(M) \mapsto \Gamma(N M)$ of an embedded Riemannian submanifold is the map (read “two”) defined by $\text{II}(X, Y) = (\tilde{\nabla}_X Y)^\perp$.
Gauss's Theorema Egregium (绝妙定理).
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 $\mathbb{R}^{2m}$.
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.