Calculus can be defined on smooth manifolds.

**Vector field** (向量场) $X: M \mapsto T M$ on a smooth manifold $M$
is a continuous map that takes each point on the manifold
to an element of the corresponding fiber of the tangent bundle: $X_p \in T_p M$.
In other words, a vector field is a section of (the projection of) the tangent bundle:
$\pi \circ X = \text{Id}_M$.
We visualize a vector field as such:
it attaches an arrow to each point of the manifold, which varies continuously across the manifold.
**Rough vector field** is almost a vector field except that it is not necessarily continuous.
**Smooth vector field** is a vector field that is a smooth map.
**Space of smooth vector fields** $\mathfrak{X}(M)$ on a smooth manifold
is the set of all smooth vector fields on the manifold
endowed with pointwise addition and scalar multiplication,
which is a real vector space and a module over $C^\infty(M)$:
$X \in \mathfrak{X}(M)$, $f \in C^\infty(M)$, then $f X \in \mathfrak{X}(M)$.
**Vector field along a subset** of a smooth manifold is a vector field on the subset.
**Smooth vector field along a subset** of a smooth manifold is a vector field along the subset
that can be smoothly extended at each point to a neighborhood in the manifold.
Any smooth vector field $X$ along a closed subset $A$ of a smooth manifold $M$
can be extended to a smooth vector field $\tilde{X}$ on $M$
that vanishes on any open subset $U$ containing $A$.

Vector fields $(X_i)_{i=1}^k$ on a subset of a smooth manifold are **linearly independent**
if they are linearly independent in the tangent space at each point;
they **span the tangent bundle** if they span the tangent space at each point.
Local **frame** $(e_i)_{i=1}^n$, or $(e_i)$, on an open subset $U$ of a smooth n-manifold $M$
is an n-tuple of vector fields on $U$ that are linearly independent and span the tangent bundle.
**Smooth frame** is a frame consisting of smooth vector fields.
**Orthonormal vector fields** on an open subset of a Euclidean space
are vector fields whose values at each point are orthonormal w.r.t. the Euclidean inner product.
**Orthonormal frame** on an open subset of a Euclidean space
is a frame consisting of orthonormal vector fields.
*Gram-Schmidt Algorithm for Frames*:
A smooth orthonormal frame $(e_j)$ can be contructed
from a smooth frame $(X_j)$ on an open subset $U$ of $\mathbb{R}^n$ such that
$\text{Span}\{e_i\}_{i=1}^j = \text{Span}\{X_i\}_{i=1}^j$ for all $j$ at each point.
**Global frame** for a smooth manifold is a frame on the entire manifold.
**Parallelizable manifold** is a smooth manifold that admits a smooth global frame.
Most smooth manifolds do not admit a smooth global frame, e.g. the sphere $\mathbb{S}^2$,
and therefore are not parallelizable.

**Coordinate vector field** $\partial / \partial x^i$ w.r.t. a smooth chart on a smooth manifold
is the vector field consisting of the i-th coordinate vector $\partial / \partial x^i |_p$
at each point of the coordinate domain.
Coordinate vector fields are smooth vector fields.
**Coordinate frame** $(\partial / \partial x^i)$
is the smooth local frame consisting of the coordinate vector fields.
**Component function** $X^i: U \mapsto \mathbb{R}$ of a rough vector field in a smooth chart
is the real-valued function on the coordinate domain that provides
the i-th component of the field w.r.t. the coordinate frame associated with the chart:
$X_p = X^i(p) \frac{\partial}{\partial x^i} \bigg{|}_p$.
The restriction of a rough vector field to the coordinate domain of a smooth chart is smooth
if and only if its component functions w.r.t. this chart are smooth.

Applying a smooth vector field $X \in \mathfrak{X}(M)$
to a smooth real-valued function $f \in C^\infty(U)$ on an open subset $U$ of $M$
is a smooth real-valued function $X f \in C^\infty(U)$, defined by $(X f)(p) = X_p f$.
For a rough vector field $X: M \mapsto T M$, the following are equivalent:
(1) $X$ is a smooth map;
(2) $X$ is a closed operator on smooth functions;
(3) $X$ is a closed operator on smooth functions on every open subset.
**Derivation** on smooth real-valued functions
is a linear transformation over $\mathbb{R}$ that satisfies the product rule:
$D \in \mathcal{L}(C^\infty(M), C^\infty(M))$;
$\forall f, g \in C^\infty(M)$, $D (f g) = (D f) g + f (D g)$.
A transformation on smooth real-valued functions on a smooth manifold is a derivation
if and only if it is a smooth vector field on the manifold.

Given a smooth map $F \in C^\infty(M, N)$,
a vector field $X$ on $M$ and a vector field $Y$ on $N$ are $F$**-related**
if $Y$ equals the pushforward of $X$ by $F$: $dF_p(X_p) = Y_{F(p)}$.
Given a smooth map $F \in C^\infty(M, N)$
and smooth vector fields $X \in \mathfrak{X}(M)$, $Y \in \mathfrak{X}(N)$,
the following are equivalent:
(1) $X$ and $Y$ are $F$-related;
(2) $X (f \circ F) = (Y f) \circ F$ for all $f \in C^\infty(U)$ on every open subset $U$ of $N$;
(3) the composition of $F$ with any integral curve of $X$ is an integral curve of $Y$.
**Pushforward** $F_∗ X$ of a smooth vector field $X \in \mathfrak{X}(M)$ by a diffeomorphism
$F: M \mapsto N$ is the smooth vector field on $N$ that is $F$-related to $X$, defined by
$(F_∗ X)_q = dF_{F^{-1}(q)}(X_{F^{-1}(q)})$.
The pushforward of $X$ by $F$ is the only smooth vector field on $N$ that is $F$-related to $X$.
A vector field $X$ on a smooth manifold $M$ is **tangent to** a smooth submanifold $S$
if it lies in the tangent subspace $T_p S$ at every point $p \in S$.
A smooth vector field $X$ on a smooth manifold $M$ is tangent to an embedded submanifold $S$
if and only if applying $X$ to any smooth real-valued function on $M$ that equals zero on $S$
gives a function that also equals zero on $S$:
$f \in C^\infty(M)$, $f|_S = 0$, then $(X f)|_S = 0$.
*Restricting Vector Fields to Submanifolds*:
If a smooth vector field $Y$ on a smooth manifold $M$ is tangent to a smooth submanifold $S$,
then the restricted vector field $Y|_S$ is the only smooth vector field on $S$
that is $\iota$-related to $Y$, where $\iota: S \mapsto M$ is the inclusion map.

Parameterized **curve** $\gamma: J \mapsto M$ in a manifold $M$
is a continuous map from an interval $J \subset \mathbb{R}$ to the manifold.
The range of a smooth parameterized curve in a smooth manifold need not be a 1-submanifold,
e.g. if a curve crosses itself, the subspace topology on the curve fails to be a manifold topology.
**Starting point** $\gamma(0)$ of a curve $\gamma$ is its value at $t = 0$ if $0 \in J$.
**Curve segment** is a curve whose domain is a compact interval: $J = [a, b]$.
**Starting point** $\gamma(a)$ and **ending point** $\gamma(b)$ of a curve segment
are the values of the ends of its domain.
**Closed curve segment** is a curve segment with identical end points: $\gamma(a) = \gamma(b)$.
**Smooth curve** in a smooth manifold is a curve that is a smooth map.
**Velocity** $\gamma'(t)$ or $\dot{\gamma}(t)$ of a smooth curve at a time instance
is the pushforward of the coordinate vector $d/dt|_t$ by the curve:
$\gamma'(t) = d \gamma_t \left(\frac{d}{d t} \bigg{|}_t \right)$.
Velocity $\gamma'(t)$ is a tangent vector of $M$ at $\gamma(t)$.
**Regular curve** in a smooth manifold is a smooth curve with nonzero velocities.
**Piecewise regular curve segment**, or "**admissible curve**" for short, in a smooth manifold
is a curve segment that can be partitioned into regular curve segments.
Any two points of a connected smooth manifold can be connected by a piecewise regular curve segement.

**Integral curve** $\gamma: J \mapsto M$ of a vector field $V$ on a smooth manifold $M$
is a differentiable curve whose velocity equals the vector field everywhere on the curve:
$\forall t \in J$, $\gamma'(t) = V_{\gamma(t)}$.
The local coordinate representation of integral curves in a smooth chart is equivalent to
the solutions of the system of ordinary differential equations (ODEs)
$\dot{\gamma}^i(t) = V^i (\gamma^i(t))_{i=1}^n$,
which is why such curves are called "integral curves".
**Maximal integral curve** is an integral curve
that cannot be extended to an integral curve on a larger open interval.

**Global flow** $\theta: \mathbb{R} \times M \mapsto M$ on a smooth manifold $M$
is a continuous map such that $\forall s, t \in \mathbb{R}$, $\forall p \in M$,
$\theta(0, p) = p$, $\theta(t, \theta(s, p)) = \theta(s+t, p)$.
Equivalently, a global flow is a continuous left $\mathbb{R}$-action on $M$,
aka a "one-parameter group action".
Every global flow induces a family $(\theta_t)_{t \in \mathbb{R}}$ of transformations on $M$
by $\forall p \in M$, $\theta_t(p) = \theta(t, p)$, and a family $(\theta^{(p)})_{p \in M}$
of curves in $M$ by $\forall t \in \mathbb{R}$, $\theta^{(p)}(t) = \theta(t, p)$.
Every transformation induced by a global flow is a homeomorphism;
it is a diffeomorphism if the global flow is a smooth map.
**Flow domain** $\mathscr{D}$ for a smooth manifold $M$ is a subset of $\mathbb{R} \times M$
such that for each $p \in M$ the subset $\mathscr{D}^{(p)} = \{t: (t, p) \in \mathscr{D}\}$
is an open interval containing zero.
Flow domains looks like open tubes around $\{0\} \times M$.
Local **flow** (流) $\theta: \mathscr{D} \mapsto M$ on a smooth manifold $M$
is a continuous map from a flow domain to the manifold such that
$\forall p \in M$, $\forall s \in \mathscr{D}^{(p)}$,
$\forall t \in \mathscr{D}^{(\theta(s, p))} \cap (\mathscr{D}^{(p)} -s)$,
$\theta(0, p) = p$, $\theta(t, \theta(s, p)) = \theta(s+t, p)$.

**Maximal flow** is a flow that that cannot be extended to a flow on a larger flow domain.
**Infinitesimal generator** $V$ of a smooth flow $\theta$ on $M$,
is the rough vector field on $M$ defined by $V_p = \theta^{(p)'}(0)$.
The infinitesimal generator $V$ of $\theta$ is a smooth vector field on $M$,
and each curve $\theta^{(p)}$ is an integral curve of $V$.
**Flow generated by** a smooth vector field
is a smooth maximal flow, if exists, whose infinitesimal generator is the field.
*Fundamental theorem on flows*:
Every smooth vector field $V$ on a smooth manifold $M$ (tangent to the boundary)
generates a unique smooth maximal flow $\theta$.
The curve $\theta^{(p)}: \mathscr{D}^{(p)} \mapsto M$
is the unique maximal integral curve of $V$ starting at each $p \in M$.
If $s \in \mathscr{D}^{(p)}$, then $\mathscr{D}^{(\theta(s, p))} = \mathscr{D}^{(p)} - s$.
For all $t \in \mathbb{R}$, $M_t = \{p: (t, p) \in \mathscr{D}\}$ is an open subset of $M$,
and $\theta_t: M_t \mapsto M_{-t}$ is a diffeomorphism with inverse $\theta_{-t}$.
**Complete vector field** is a smooth vector field that generates a global flow.
Every compactly-supported smooth vector field on a smooth manifold is complete.
Every smooth vector field on a compact smooth manifold is complete.

*Flowout Theorem*:
Given an embedded submanifold $S$ of a smooth manifold $M$
and a smooth vector field $V$ on $M$ that is nowhere tangent to $S$,
let $V$ generates flow $\theta$ with flow domain $\mathscr{D}$,
denote restricted flow domains $\mathscr{O} = \{(t, p) \in \mathscr{D}: p \in S\}$
and $\mathscr{O}_\delta = \{(t, p) \in \mathscr{O} : |t| < \delta(p)\}$,
where $\delta$ is a smooth positive function on $S$, then:
(1) the restricted flow $\theta|_\mathscr{O}$ is a smooth immersion;
(2) the coordinate vector field $\partial / \partial t$ on $\mathscr{O}$
is $\theta|_\mathscr{O}$-related to $V$;
(3) the restricted flow $\theta|_{\mathscr{O}_\delta}$ can be injective for some $\delta$,
and thus its range $\theta(\mathscr{O}_\delta)$---called a **flowout** (流出) from $S$ along $V$---
is an immersed submanifold of $M$ containing $S$, and $V$ is tangent to this submanifold;
(4) if $S$ has codimension one, then the restricted flow $\theta|_{\mathscr{O}_\delta}$
is a diffeomorphism onto the flowout, which is an open submanifold of $M$.

**Equilibrium point** of a flow $\theta: \mathscr{D} \mapsto M$ on a smooth manifold $M$
is a point $p$ in $M$ such that $\forall t \in \mathscr{D}^{(p)}$, $\theta(t, p) = p$.
**Singular point** (奇点) or **zero** of a vector field on a smooth manifold
is a point where the vector field is zero: $V_p = 0$.
**Regular point** (常点) of a vector field on a smooth manifold
is a point where the field is nonzero.
The singular points of a smooth vector field are precisely
the equilibrium points of the flow it generates.
*Canonical Form Near a Regular Point*:
A smooth vector field $V$ matches the first coordinate vector field
w.r.t. a smooth chart on a neighborhood $U$ of any regular point $p$,
and the first coordinate can be a local defining function
for any embedded hypersurface $S$ containing $p$ given that $V$ is not tangent to $S$ at $p$:
$V|_U = \partial/\partial x^1$, $S \cap U = (x^1)^{-1}(0)$.

Real-valued first-order partial differential equations (PDEs)
can be reduced to ODEs by the theory of flows.
**Linear first-order Cauchy problem** is a problem of finding a smooth real-valued function $u$
in a neighborhood of an embedded hypersurface $S$ in a smooth manifold $M$
that satisfies a linear first-order PDE $A u + b u = f$ and an initial condition $u|_S = \phi$,
where $A$ is a smooth vector field on $M$, $b$ and $f$ are smooth real-valued functions on $M$,
and $\phi$ is a smooth real-valued function on $S$.
**Characteristic line** (特征线)...
A linear first-order Cauchy problem is **noncharacteristic** if $A$ is nowhere tangent to $S$.
If a linear first-order Cauchy problem is noncharacteristic,
then it has a unique solution in a flowout from the initial hypersurface along the vector field.
Given a restricted flow domain $\mathscr{O}_\delta$ that satisfies the Flowout Theorem,
composition with the restricted flow $\theta_\delta := \theta|_{\mathscr{O}_\delta}$ of $A$
transforms the flowout to the restricted flow domain
where $A$ is in its canonical form $\partial/\partial t$,
so the PDE becomes a linear first-order ODE
$\frac{\partial \hat{u}}{\partial t} + \hat{b} \hat{u} = \hat{f}$
with initial condition $\hat{u}(0) = \phi$,
where $\hat{u} = u \circ \theta_\delta$, and $\hat{b}, \hat{f}$ are similarly defined.
Thus the solution in the flowout is $u = \hat{u} \circ \theta_\delta^{-1}$,
where $\hat{u}(t) = e^{-B(t)} \left(\phi + \int_0^t \hat{f}(\tau) e^{B(\tau)}~d \tau\right)$
and $B(t) = \int_0^t \hat{b}(\tau)~d\tau$.

**1-jet bundle** $J^1 M$ of a smooth manifold $M$
is the smooth vector bundle $J^1 M = \mathbb{R} \times T^∗ M \mapsto M$,
with fibers $\mathbb{R} \times T_x^∗ M$.
**1-jet** $j^1 u$ of a smooth function $u \in C^\infty(M)$
is the section of the 1-jet bundle $J^1 M$ defined by $j^1 u = (u, d u)$.
**First-order Cauchy problem** is a problem of finding a real-valued function $u$
in a neighborhood of an embedded hypersurface $S$ in a smooth manifold $M$
that satisfies a first-order PDE $F(x, u, d u) = 0$ and an initial condition $u|_S = \phi$,
where $\phi$ is a smooth real-valued function on $S$ and
$F$ is a smooth real-valued function on an open subset of the 1-jet bundle $J^1 M$.
A first-order Cauchy problem is **noncharacteristic**
if there is a smooth section $\sigma$ of $T^∗ M|_S$ that takes values in $W$ and satisfies
$\sigma(x)|_{T_x S} = d \phi(x)$ and $F(x, \phi(x), \sigma(x)) = 0$ on all points $x$ in $S$,
and the vector field $A^{\phi, \sigma}$ along $S$ is nowhere tangent to $S$, defined as
$A^{\phi, \sigma}|_x = \sum_{i=1}^n
\frac{\partial F}{\partial \xi_i}(x, \phi(x), \sigma(x)) \frac{\partial}{\partial x^i}$.
If a first-order Cauchy problem is noncharacteristic,
then it has a smooth solution on a neighborhood of each point on the hypersurface.

Integral manifold is a generalization of integral curve to higher-dimensional submanifolds.

Rank-k **distribution**, **tangent distribution**, **tangent subbundle**, or **k-plane fields** $D$
on a smooth manifold is a k-dimensional subbundle of the tangent bundle:
$D = \sqcup_{p \in M} D_p$, $D_p \in G_k(T_p M)$.
**Smooth distribution** on a smooth manifold is a distribution that is a smooth subbundle,
i.e. locally spanned by k-tuples of smooth vector fields:
$\forall p \in M$, $\exists U \subset M$, $\exists (X_i)_{i=1}^k \subset \mathfrak{X}(U)$:
$\forall q \in U$, $\text{span}(X_{i,q})_{i=1}^k = D_q$.
We denote the space of smooth global sections of a smooth distribution as $\Gamma(D)$;
note that $\Gamma(D) \subset \mathfrak{X}(M)$.
**Involutive distribution** on a smooth manifold is a smooth distribution such that
the Lie bracket of any pair of its smooth local sections is also a local section; or equivalently,
the space $\Gamma(D)$ of its smooth global sections is a Lie subalgebra of $\mathfrak{X}(M)$.
*Local Frame Criterion for Involutivity*:
A distribution is involutive if the manifold can be covered by smooth local frames such that
the Lie bracket of every pair of vector fields in a frame is a local section of the distribution.
Involutivity can be rephrased in terms of differential forms.

**Integral manifold** of a smooth rank-k distribution on a smooth manifold
is an immersed k-submanifold whose tangent space at each point matches the distribution:
$N \subset M$, $\forall p \in N$, $T_p N = D_p$.
**Integrable distribution** on a smooth manifold is a smooth distribution such that
each point of the manifold is contained in an integral manifold of the distribution.
Every integrable distribution is involutive.
**Flat chart** for a rank-k distribution on a smooth n-manifold
is a smooth coordinate chart whose first k coordinate vector fields span the distribution,
and whose image is a box.
In a flat chart, the preimage of every slice with fixed last (n-k) coordinates
is an integral manifold of the distribution.
**Completely integrable distribution** on a smooth manifold is a smooth distribution such that
the manifold can be covered by flat charts for the distribution.
Every completely integrable distribution is integrable.
*Frobenius theorem*:
Every involutive distribution is completely integrable.
Therefore, a distribution is (completely) integrable if and only if it is involutive.
*Local structure of integral manifolds*:
For an involutive rank-k distribution on a smooth manifold,
the intersection of any integral manifold and the coordinate domain of a flat chart
is a countable union of disjoint open subsets of parallel k-dimensional slices,
each of which is an open subset of the integral manifold and an embedded submanifold.
**Weakly embedded submanifold** in a smooth manifold is a smooth submanifold such that
every smooth map whose image lies in the submanifold is smooth as a map to the submanifold:
$H \subset M$, $\forall F \in C^\infty(N, M)$, $F(N) \subset H \implies F \in C^\infty(N, H)$.
Every integral manifold of an involutive distribution is weakly embedded.

**Flat chart** for a collection of k-submanifolds of a smooth n-manifold
is a smooth coordinate chart whose image is a box,
and every submanifold in the collection intersects the coordinate domain in either the empty set
or a countable union of preimages of slices with fixed last (n-k) coordinates.
**Foliation** (叶状结构) $\mathscr{F}$ of dimension k on a smooth n-manifold
is a partition of the manifold into connected, nonempty, immersed k-submanifolds
(called the **leaves** $L$ of the foliation), for which there are flat charts covering the manifold.
The collection of tangent spaces to the leaves of a foliation on a smooth manifold
forms an involutive distribution on the manifold.
*Global Frobenius Theorem*:
The collection of all maximal connected integral manifolds
of an involutive distribution on a smooth manifold forms a foliation of the manifold.
Therefore, foliations are in one-to-one correspondence with involutive distributions.
**Invariant distribution** on a smooth manifold w.r.t. a diffeomorphic transformation,
or **Φ-invariant distribution**, is a distribution that is invariant under pushforward by the map:
$\Phi: M \cong M$, $\forall x \in M$, $d \Phi_x (D_x) = D_{\Phi(x)}$.
Similarly, **Φ-invariant foliation** on a smooth manifold
is a foliation that is invariant under the map:
$\forall L \in \mathscr{F}$, $\Phi(L) \in \mathscr{F}$.
An involutive distribution is Φ-invariant if and only if the foliation it determines is Φ-invariant.

**Overdetermined system of partial differential equations** is one
where the number of PDEs is larger than that of unknown functions.

**Overdetermined linear first-order Cauchy problem**:
given a linearly independent m-tuple of smooth vector fields $(A_i)_{i=1}^m$
on an open subset $W$ of the Euclidean n-space, $m \le n$,
and an embedded (n-m)-submanifold $S \subset W$ that is transverse to the vector fields,
find a smooth real-valued function $u$ that satisfies
equations $A_i u = f_i$ with initial condition $u|_S = \phi$,
where $(f_i)_{i=1}^m \subset C^\infty(W)$ and $\phi \in C^\infty(S)$.
An overdetermined first-order Cauchy problem
has a unique solution on a neighborhood of each point on the submanifold,
if it satisfies the following compatibility conditions:
$\exists c_{ij}^k \in C^\infty(W)$:
(involutivity) $[A_i, A_j] = c_{ij}^k A_k$, and $A_i f_j - A_j f_i = c_{ij}^k f_k$.

Consider an overdetermined system of first-order PDEs: $\nabla u(x) = J(x, u(x))$, where $u \in C^\infty(U, \mathbb{R}^m)$, $U \subset \mathbb{R}^n$, $J \in C^\infty(W, M_{m,n}(\mathbb{R}))$, $W \subset \mathbb{R}^n \times \mathbb{R}^m$. If the matrix-valued function satisfies $\forall i \in m$, $\forall j, k \in n$, $\frac{\partial J^i_j}{\partial x^k} + J^l_k \frac{\partial J^i_j}{\partial y^l} = \frac{\partial J^i_k}{\partial x^j} + J^l_j \frac{\partial J^i_k}{\partial y^l}$, then for any initial condition $u(x_0) = y_0$, $(x_0, y_0) \in W$, the PDE system has a unique solution in a neighborhood of $x_0$.

Exterior differentiation allows for a generalization of differential operators such as gradient, divergence, curl, and Laplacian.

**Covector field** $\omega$ is a local or global section of the cotangent bundle.
The value $\omega_p$ of a covector field at a point is denoted by subscript,
while parentheses are reserved for the action $\omega(v)$ of a covector on a vector.
We visualize a covector field as such:
in each tangent space, it defines a linear hyperplane as the zero set
and a parallel affine hyperplane as the level set of one,
both of which vary continuously across the manifold.
As with vector fields, a rough field needs not be continuous, and a smooth field is smooth.
**Action** $\omega(X)$ of a rough covector field on a vector field on a smooth manifold
is the real-valued function on the manifold that equals the action of the covector on the vector
at each point: $\forall p \in M$, $\omega(X)(p) = \omega_p(X_p)$.
**Space of smooth covector fields** $\mathfrak{X}^∗ (M)$ on a smooth manifold,
endowed with pointwise vector addition and scalar multiplication,
is a real vector space and a module over $C^\infty(M)$.

Local **coframe** $(\varepsilon^i)_{i=1}^n$ is a local frame for the cotangent bundle.
**Smooth coframe** is a coframe consisting of smooth covector fields.
**Global coframe** is a coframe on the entire manifold.
**Component functions** $\omega_i: U \mapsto \mathbb{R}$
of a rough covector field w.r.t. a coframe $(\varepsilon^i)$
are the maps whose values form the coordinate representation of the covector at each point:
$\omega_i(p) = \omega_p(e^i |_p)$, where $(e_i)$ is the dual frame.
Given a coframe, a covector field can be written uniquely as $\omega = \omega_i \varepsilon^i$.
**Coordinate coframe** $(\lambda^i)$ is a smooth local coframe
consisting of the coordinate covector fields associated with a smooth chart.
A coframe $(\varepsilon^i)$ and a frame $(e_i)$ are **dual** to each other if
their values at each point are dual basis: $\varepsilon^i(e_j) = \delta^i_j$.
**Component functions** of a rough covector field w.r.t. a smooth chart
are the component functions of the field w.r.t. the coordinate coframe:
$\omega_i(p) = \omega_p(\partial/\partial x^i |_p)$.
The action of a rough covector field on a vector field equals
the sum of products of their component functions in any smooth frame and its dual coframe:
$\omega(X)(p) = \omega_i X^i$.

**Pullback** $F^∗ \omega$ of a covector field $\omega$ on $N$ by a smooth map $F \in C^\infty(M, N)$
is the rough covector field on $M$ whose value at each point equals
the pullback of the covector field at that point: $(F^∗ \omega)_p = d F_p^∗ (\omega_{F(p)})$,
i.e. $\forall v \in T_p M$, $(F^∗ \omega)_p (v) = \omega_{F(p)}(d F_p (v))$.
The pullback of any covector field by a smooth map is a covector field;
if the covector field is smooth, its pullback is also smooth.
**Restriction** $\iota^∗ \omega$ of a smooth covector field $\omega \in \mathfrak{X}^∗ (M)$
to a smooth submanifold is the pullback of the field by the inclusion map $\iota: S \mapsto M$;
equivalently, it is the restriction of the covector field to vectors tangent to the submanifold.

**Differential form** of degree $k$ or **k-form** $\omega$
is a section of the alternating k-tensor bundle,
i.e. an alternating k-tensor field, aka a k-covector field.
**Space of smooth k-forms** $\Omega^k(M)$ on a smooth manifold
is the space of smooth alternating k-tensor fields: $\Omega^k(M) = \Gamma(\Lambda^k T^∗ M)$.
The space of smooth 1-forms is just the space of smooth covector fields:
$\Omega^1(M) = \mathfrak{X}^∗ (M)$.
Sum **space of smooth differential forms** $\Omega^∗ (M)$ on a smooth n-manifold
is the direct sum of all the smooth k-form spaces on the manifold:
$\Omega^∗ (M) = \oplus_{k=0}^n \Omega^k(M)$.
Exterior algebra $(\Omega^∗ (M), \wedge)$ of a smooth n-manifold $M$
is the associative, anticommutative graded algebra
consisting of its space of smooth differential forms and the pointwise wedge product.

**Component function** $\omega_I$ of a rough k-form w.r.t. a smooth chart
is the action of the k-form on the k-tuple of coordinate vector fields
indexed by an increasing multi-index:
$\omega_I = \omega(\partial/\partial x^i)_{i \in I}$.
Given a smooth chart, every k-form can be written uniquely as a linear combintion of
elementary k-forms based on the coordinate coframe and increasing multi-indices of length $k$:
$\omega = \sum_I' \omega_I d x^I$, where $d x^I = \wedge_{i \in I} d x^i$.

**Pullback** $F^∗ \omega$ of a k-form on $N$ by a smooth map $F \in C^\infty(M, N)$
is the rough k-form on $M$ whose value at each point equals the pullback of the k-covector
at that point: $(F^∗ \omega)_p = d F_p^∗ (\omega_{F(p)})$, i.e. $\forall v_i \in T_p M$,
$(F^∗ \omega)_p (v_i)_{i=1}^k = \omega_{F(p)}(d F_p (v_i))_{i=1}^k$.
The pullback of any k-form by a smooth map is a k-form;
if the k-form is smooth, its pullback is also smooth.
Given a smooth chart $(y^i)$ on the codomain,
the pullback of a k-form by a smooth map can be written as:
$F^∗ (\sum_I' \omega_I d y^I) = \sum_I' (\omega_I \circ F) \bigwedge_{i \in I} d (y^i \circ F)$.
Given a smooth chart $(x^i)$ on the domain and a smooth chart $(y^i)$ on the codomain,
the pullback of an n-form by a smooth map can be written as:
$F^∗ (u (\wedge_{i=1}^n d y^i)) = (u \circ F) (\det DF) (\wedge_{i=1}^n d y^i)$,
where $\det DF$ is the determinant of the Jacobian matrix of the map in these coordinates.

The most important application of covector field is to allow for
an invariant definition of the differential of a smooth real-valued function on a smooth manifold.
**Differential** (微分) $d f$ of a smooth real-valued function on a smooth manifold
is the covector field defined by $\forall p \in M$, $\forall v \in T_p M$, $d f_p(v) = v f$.
Due to the canonical identification $T_p \mathbb{R} \leftrightarrow \mathbb{R}$,
the definitions of the differential of a smooth real-valued function
as a tangent map $d f: T M \mapsto T \mathbb{R}$ and
as a covector field where $d f_p: T_p M \mapsto \mathbb{R}$ are the same.
The action of the differential of a smooth real-valued function on a vector field
is thus $d f(X) = X f$.
The differential of a smooth function is a smooth covector field:
$d: C^\infty(M) \mapsto \mathfrak{X}^∗ (M)$.
The component functions of a differential in a smooth chart
are the partial derivatives w.r.t. those coordinates:
$d f = \frac{\partial f}{\partial x^i} \lambda^i$.
The differential of a coordinate function is the corresponding coordinate covector field:
$d x^i = \lambda^i$; we therefore use $d x^i$ to denote a coordinate covector field.

**Map of degree m** on a graded algebra $A = \oplus_k A^k$
is a linear transformation that maps each subspace to the subspace $m$ indices higher.
**Antiderivation** on a graded algebra is a linear transformation such that
$T (x \times y) = (T x) \times y + (-1)^k x \times (T y)$ where $x \in A^k$.
**Exterior differentiation** (外微分) $d: \Omega^* (M) \mapsto \Omega^* (M)$ of smooth forms
is the unique extension of the differential $d: C^\infty(M) \mapsto \mathfrak{X}^* (M)$
to an antiderivation of degree +1 on the exterior algebra whose square is zero.
Exterior differentiation has the following properties:
(1) map of degree +1: $\forall k \in \{i\}_{i=0}^n$,
$d \in \mathcal{L}(\Omega^k(M), \Omega^{k+1}(M))$;
(2) antiderivation: $d(\omega \wedge \eta) = d \omega \wedge \eta + (-1)^k \omega \wedge d \eta$,
where $\omega \in \Omega^k(M)$;
(3) repeated action vanishes: $d \circ d = 0$.
(4) commutes with pullbacks: $\forall F \in C^\infty(M, N)$, $\forall \omega \in \Omega^* (N)$,
$F^* (d \omega) = d(F^* \omega)$.
**Exterior derivative** (外导数) $d \omega$ of a smooth k-form
on an open submanifold or a regular domain of a Euclidean space
is the (k+1)-form defined by $d(\sum_I' \omega_I d x^I) = \sum_I' d \omega_I \wedge d x^I$,
where $(d x^i)$ is the standard coordinate coframe.
In particular, the exterior derivative of a smooth 1-form can be written as:
$d(\omega_j d x^j) = \sum_{i<j} \left( \frac{\partial \omega_j}{\partial x^i} -
\frac{\partial \omega_i}{\partial x^j} \right) d x^i \wedge d x^j$.
Given a smooth chart, the exterior differentiation can be written in the form of
the exterior derivative, where $(d x^i)$ is the coordinate coframe.

**Exact covector field** or **exact differential**
is a smooth covector field that equals the differential of a smooth real-valued function:
$\exists f \in C^\infty(M)$, $\omega = d f$.
We call this function a **potential** for the exact covector field.
The potentials for an exact covector field differ only by a constant
on each component of the manifold.
**Conservative covector field** is a smooth covector field
whose line integral over every piecewise smooth closed curve segment is zero;
equivalently, its line integrals over piecewise smooth curve segments are path-independent,
i.e. only depend on the starting and ending points.
A smooth covector field is conservative if and only if it is exact.
**Closed covector field** is a smooth covector field
whose Jacobian in every smooth chart is symmetric, or equivalently,
whose Jacobian in every chart in a smooth atlas is symmetric:
$\frac{\partial \omega_j}{\partial x^i} = \frac{\partial \omega_i}{\partial x^j}$.
Every exact covector field is closed.
The pullback of a covector field by a local diffeomorphism
preserves closedness and exactness of the covector field.
**Star-shaped subset** of a vector space
is a subset that includes the line segment between one point and any point in the subset:
$\exists c \in A$, $A = \cup_{p \in A} \overline{cp}$.
Every convex subset is star-shaped.
*Poincaré Lemma for Covector Fields*:
Every closed covector field on a star-shaped open subset of
a Euclidean space $\mathbb{R}^n$ or a closed upper half-space $\mathbb{H}^n$ is exact.
Every closed covector field is exact on a collection of open sets that cover the manifold.
Every closed covector field is exact on any simply connected manifold.
**Exact k-form** is a k-form that equals the exterior differentiation of a smooth (k-1)-form:
$\exists \eta \in \Omega^{k-1}(M)$, $\omega = d \eta$.
**Closed k-form** is a smooth k-form whose exterior differentiation is zero: $d \omega = 0$.
Every exact differential form is closed.
Every closed differential form is locally exact.

Figure: Commutative diagram of some differential operators on an oriented Riemannian 3-manifold.

Every pseudo-Riemannian metric $g$ is equivalent to a smooth bundle isomorphism
$\hat{g}: T M \mapsto T^∗ M$ defined by $\hat{g}(v)(w) = g_p(v, w)$.
**Musical isomorphisms** between smooth vector fields and smooth covector fields
on a pseudo-Riemannian manifold $(M, g)$ are the two vector space (and module) isomorphisms
**flat** (降X) or **lower an index** (降指标) $\flat: \mathfrak{X}(M) \mapsto \mathfrak{X}^∗ (M)$ and
**sharp** (升ω) or **raise an index** (升指标) $\sharp: \mathfrak{X}^∗ (M) \mapsto \mathfrak{X}(M)$
defined by $X^\flat (Y) = \hat{g}(X)(Y) = g(X, Y)$ and $\omega^\sharp = \hat{g}^{-1}(\omega)$.
Given a smooth local frame $(e_i)$ and its dual coframe $(\varepsilon^i)$,
the musical isomorphisms have coordinate representations
$X^\flat = g_{ij} X^i \varepsilon^j$ and $\omega^\sharp = g^{ij} \omega_i e_j$,
where $(g^{ij})$ is the inverse of the matrix representation of the pseudo-Riemannian metric
so that $g^{ij} g_{jk} = \delta^i_k$.

**Gradient** $\text{grad}~f$ of a smooth real-valued function on a pseudo-Riemannian manifold
is the vector field obtained from the differential of the function by raising an index:
$\text{grad}~f = (d f)^\sharp$.
Gradient and differential on a Riemannian manifold are related by
$d f_p = \langle \text{grad}~f|_p, \cdot \rangle_g$.
Given a smooth local frame, gradient can be written as $\text{grad}~f = g^{ij} (e_i f) e_j$.
Gradient and differential have the same coordinate representation in any orthonormal frame.

**Divergence** $\text{div}~X$ of a smooth vector field on a Riemannian n-manifold $(M, g)$
is the smooth real-valued function that locally satisfies the equation
$d(X \lrcorner d V_g) = (\text{div}~X) d V_g$, where $d V_g$ is the Riemannian density.
Given a smooth coordinate frame, divergence can be written as
$\text{div}~X = (\sqrt{\det g})^{-1} \frac{\partial}{\partial x^i} (X^i \sqrt{\det g})$,
where $\det g$ the determinant of the component matrix of the Riemannian metric in these coordinates.

**Curl** $\text{curl}~X$ of a smooth vector field on an oriented Riemannian 3-manifold
is the smooth 2-form defined by $\text{curl}~X = \beta^{-1} d(X^\flat)$,
where $\beta: TM \mapsto \Lambda^2 T^* M$ is the smooth bundle isomorphism defined by
$\beta(X) = X \lrcorner d V_g$.

Geometric **Laplacian** or **Laplace–Beltrami operator** $\Delta f$
of a smooth real-valued function on a Riemannian manifold
is the smooth real-valued function defined by the divergence of the gradient of the function:
$\Delta f = \text{div}(\text{grad}~f)$.
Many authors define the Laplacian with a negative sign so that its eigenvalues are nonnegative,
but the given definition is much more common in Riemannian geometry.
Given a smooth coordinate frame, Laplacian can be written as
$\Delta f = (\sqrt{\det g})^{-1} \frac{\partial}{\partial x^i}
\left( g^{ij} \frac{\partial f}{\partial x^j} \sqrt{\det g} \right)$.

**Hodge star operator** $∗$ is the smooth bundle homomorphism between alternating tensor bundles
$\Lambda^k T^∗ M$ and $\Lambda^{n-k} T^∗ M$ on an oriented Riemannian n-manifold $(M, g)$
for each $k \in \{i\}_{i=0}^n$, determined by $\forall \omega, \eta \in \Omega^k(M)$,
$\omega \wedge ∗ \eta = \langle \omega, \eta \rangle_g d V_g$.
In particular, for smooth real-valued functions, $∗ f = f d V_g$.
**Laplace–Beltrami operator** $\Delta \omega$
of a smooth k-form on an oriented compact Riemannian n-manifold
is the smooth k-form defined by $\Delta \omega = d d^∗ \omega + d^∗ d \omega$,
where $d^∗$ is the map of degree -1 defined by $d^∗ \omega = (-1)^{n(k+1)+1} ∗ d ∗ \omega$,
where $∗$ is the Hodge star operator.
**Harmonic k-form** is a smooth k-form in the kernel of the Laplace–Beltrami operator:
$\Delta \omega = 0$.
**Harmonic function** is a Harmonic 0-form.
Harmonic analysis of real-valued functions on smooth manifolds, e.g. spherical harmonics.

Covariant differentiation, or connection, allows for a generalization of differential operators such as directional derivative and Hessian.

Connection is a rule that "connects" nearby tangent spaces on a smooth manifold,
such that tangent vectors at different points can be compared,
and directional derivatives of vector fields can be computed intrinsically.
**Connection** $\nabla$ in a smooth vector bundle over a smooth manifold
is a map that takes a smooth vector field on the manifold and a smooth section of the bundle
to another smooth section of the bundle, which is linear over smooth functions in the first argument,
is linear over real numbers in the second, and satisfies the product rule:
give $(E, \pi)$ over $M$, $\nabla: \mathfrak{X}(M) \times \Gamma(E) \mapsto \Gamma(E)$:
(1) $\forall f_1, f_2 \in C^\infty(M)$,
$\nabla_{f_1 X_1 + f_2 X_2} Y = f_1 \nabla_{X_1} Y + f_2 \nabla_{X_2} Y$;
(2) $\forall a_1, a_2 \in \mathbb{R}$,
$\nabla_X (a_1 Y_1 + a_2 Y_2) = a_1 \nabla_X Y_1 + a_2 \nabla_X Y_2$;
(3) $\forall f \in C^\infty(M)$, $\nabla_X (f Y) = f \nabla_X Y + (X f) Y$.
A connection in the tangent bundle of a smooth manifold
is often simply called a "connection on the manifold":
$\nabla: \mathfrak{X}(M) \times \mathfrak{X}(M) \mapsto \mathfrak{X}(M)$.
*Existence of connections*:
The tangent bundle of every smooth manifold admits a connection.
**Covariant derivative** $\nabla_X Y$ (or "invariant derivative") of $Y$ in the direction of $X$,
w.r.t. a connection $\nabla$, where $X \in \mathfrak{X}(M)$ is a smooth vector field and
$Y \in \Gamma(E)$ is a smooth section of a vector bundle,
is the smooth section of the vector bundle given by the connection.

**Connection coefficients** $\Gamma_{ij}^k$ of a connection on a smooth manifold,
w.r.t. a smooth local frame, are the smooth real-valued functions that provide the k-th coordinate
of the covariant derivative of the j-th vector field in the direction of the i-th vector field:
$\forall i, j \in n$, $\nabla_{E_i} E_j = \Gamma_{ij}^k E_k$.
Given a smooth local frame over an open subset of a smooth manifold,
a connection on the manifold is completely determined in the subset by its connection coefficients:
$\forall X, Y \in \mathfrak{X}(U)$, let $X = X^i E_i$, $Y = Y^j E_j$,
then $\nabla_{X} Y = (X(Y^k) + X^i Y^j \Gamma_{ij}^k) E_k$.
This formula gives a bijection between connections and $n^3$-tuples of smooth real-valued functions
on a coordinate domain of a smooth manifold;
for a smooth n-manifold with a global frame, the set of connections on its tangent bundle
is equipotent to the $n^3$-power of smooth real-valued functions on the manifold.

**Euclidean directional derivative** $\bar{\nabla}_X Y$
of a smooth vector field by a vector field in the Euclidean n-space is the vector field defined by:
$\bar{\nabla}_X Y = (\nabla Y) X$, where $X$ and $Y$ on the right hand side
are understood as their component functions in the standard coordinates.
**Euclidean connection** $\bar{\nabla}$ on a Euclidean space
is the connection in the Euclidean tangent bundle that provides the Euclidean directional derivative:
$\bar{\nabla}_X Y = X(Y^k) \frac{\partial}{\partial x^k}$.
The connection coefficients of $\bar{\nabla}$ in the standard coordinate frame are all zero:
$\bar \nabla_{E_i} E_j = 0$.
**Tangential directional derivative** $\nabla^\top_v Y$
of a smooth vector field in an embedded submanifold of a Euclidean space in a tangent direction
is the directional derivative of a smooth extension of the vector field in this direction,
projected to the tangent space at each point:
$\nabla^\top_v Y = \pi^\top(\bar{\nabla}_v \tilde Y)$, where $\tilde Y |_M = Y$.
**Tangential connection** $\nabla^\top$ on an embedded submanifold of a Euclidean space
is the connection that provides the tangential directional derivative:
$\forall X, Y \in \mathfrak{X}(M)$, $\tilde X|_M = X$, $\tilde Y|_M = Y$,
$\nabla^\top_X Y = \pi^\top (\bar \nabla_{\tilde X} \tilde Y)$.

Every connection in the tangent bundle induces connections in all tensor bundles,
and thus defines covariant derivatives of tensor fields of any type.
**Total covariant derivative** $\nabla F$ of a smooth (k,l)-tensor field on a smooth manifold,
given a connection in the tangent bundle, is the smooth (k,l+1)-tensor field on the manifold
whose action on k smooth covector fields and l+1 smooth vector fields is the action of
the covariant derivative of the tensor field in the direction of the last vector field:
$\forall F \in \Gamma(T^{(k, l)} T M)$, $\exists \nabla F \in \Gamma(T^{(k, l+1)} T M)$:
$\forall \omega^i \in \Omega^1(M)$, $\forall X, Y_j \in \mathfrak{X}(M)$,
$(\nabla F)(\omega^i, Y_j, X)^{i \in k}_{j \in l} =
(\nabla_X F)(\omega^i, Y_j)^{i \in k}_{j \in l}$.
The components of a total covariant derivative in a local frame
are written with a semicolon to separate the new index from the preceding indices:
for a (k,l)-tensor field, its total covariant derivative have components of the form
$F_{(i_{j'})_{j'=1}^l;m}^{(i_j)_{j=1}^k}$.
The musical isomorphisms commute with the total covariant derivative operator:
$\nabla(F^\flat) = (\nabla F)^\flat$, and $\nabla(F^\sharp) = (\nabla F)^\sharp$.

The (total) covariant derivative of a smooth function on a smooth manifold
is its differential 1-form: $\forall u \in C^\infty(M)$, $\nabla u = d u$.
**Covariant Hessian** $\nabla^2 u$ of a smooth function on a smooth manifold
is the (0,2)-tensor field defined by its second (total) covariant derivative:
$\nabla^2 u = \nabla (d u)$;
its action on smooth vector fields can be computed by $\nabla^2 u (Y, X) = Y(X u) - (\nabla_Y X) u$.
**Hessian operator** $\mathcal{H}_u$ (or $\text{Hess}~u$) of a smooth function on a smooth manifold
is the (1,1)-tensor field, i.e. an endomorphism field,
obtained from the covariant Hessian by raising an index: $\mathcal{H}_u = (\nabla^2 u)^\sharp$;
or equivalently, the covariant derivative of the gradient: $\mathcal{H}_u = \nabla (\text{grad}~u)$,
because $(\nabla (\nabla u))^\sharp = (\nabla (d u))^\sharp = \nabla ((d u)^\sharp) =
\nabla (\text{grad}~u)$.

**Vector field along a smooth curve** in a smooth manifold
is a parameterized curve in the tangent bundle that is compatible with the curve:
$V \in C(I, T M)$, $\gamma \in C^\infty(I, M)$, $\forall t \in I$, $V(t) \in T_{\gamma(t)} M$.
The set $\mathfrak{X}(\gamma)$ of all smooth vector fields along a smooth curve
is a real vector space under pointwise vector addition and multiplication,
and is a module over $C^\infty(I)$ with pointwise multiplication.
The velocity of smooth curve is a vector field along the curve: $\gamma' \in \mathfrak{X}(\gamma)$.
**Extendible vector field along a smooth curve** is a smooth vector field along the curve
such that there exists a smooth vector field on a neighborhood of the image of the curve
that is compatible with this vector field:
$\exists \tilde{V} \in \mathfrak{X}(U)$, $U \supset \gamma(I)$: $V = \tilde{V} \circ \gamma$.
**Tensor field along a smooth curve** in a smooth manifold
is a parameterized curve in a tensor bundle that is compatible with the curve:
$\sigma \in C(I, T^{(k,l)} T M)$, $\gamma \in C^\infty(I, M)$,
$\forall t \in I$, $\sigma(t) \in T^{(k,l)} (T_{\gamma(t)} M)$.
**Covariant derivative** or **absolute derivative** $D_t$ along a smooth curve in a smooth manifold
is an operator on the space of smooth vector fields along the curve,
uniquely determined by a connection in the tangent bundle, which is linear over the real numbers,
satisfies the product rule, and equals the covariant derivative of every extension
of an extendible vector field in the direction of the curve's velocity:
$D_t: \mathfrak{X}(\gamma) \mapsto \mathfrak{X}(\gamma)$;
(1) $\forall a, b \in \mathbb{R}$, $D_t(a V + b W) = a D_t V + b D_t W$;
(2) $\forall f \in C^\infty(I)$, $D_t (f V) = f' V + f D_t V$;
(3) if $V$ can be extended to $\tilde V$, then $D_t V(t) = \nabla_{\gamma'(t)} \tilde V$.

Recall that for a smooth curve in a Euclidean space, its (Euclidean) velocity and acceleration
are the first and second derivatives of its component functions in the standard coordinates:
$\gamma'(t) = \frac{d \gamma}{d t}$, $\gamma''(t) = \frac{d^2 \gamma}{d t^2}$.
**Tangential acceleration** $\gamma''^\top$ of a smooth curve on an embedded submanifold
of the Euclidean n-space is its Euclidean acceleration projected to the tangent space at each point:
$\gamma \in C^\infty(I, M)$, $\forall t \in I$, $\gamma''(t)^\top = \pi^\top(\gamma''(t))$,
where $\pi^\top: T \mathbb{R}^n|_M \mapsto T M$ is the tangential projection.
**Acceleration** $D_t \gamma'$ of a smooth curve in a smooth manifold
is the covariant derivative of its velocity, w.r.t. a connection in the tangent bundle.
**Geodesic** in a smooth manifold, w.r.t. a connection in the tangent bundle,
is a smooth curve with zero acceleration:
$\gamma \in C^\infty(I, M)$, $\forall t \in I$, $D_t \gamma'(t) = 0$.
**Geodesic segment** is a geodesic whose domain is a compact interval.
*Existence and uniqueness of geodesics*:
For every tangent vector on a smooth manifold, with a connection in the tangent bundle,
there exists a geodesic with that initial velocity defined on a neighborhood of zero,
and any two such geodesics are identical on their common domain:
$\forall (p, v) \in T M$, $\exists \gamma: (a, b) \mapsto M$, $a < 0 < b$:
$\gamma(0) = p$, $\gamma'(0) = v$, and $\forall t \in (a, b)$, $D_t \gamma'(t) = 0$.
**Maximal geodesic** $\gamma_v$ with initial velocity $v$ on a smooth manifold,
w.r.t. a connection in the tangent bundle, for any tangent vector $v$,
is the geodesic whose initial location and velocity are specified by that tangent vector,
and whose domain cannot be extended to a larger interval:
$\gamma_v = \bigcup \{\gamma : \gamma(0) = p, \gamma'(0) = v, D_t \gamma'(t) = 0\}$.
**Geodesically complete** smooth manifold, w.r.t. a connection in the tangent bundle,
is one such that every maximal geodesic is defined for the entire real line:
$\forall v \in T M$, $\gamma_v: \mathbb{R} \mapsto M$.

**Parallel vector field along a smooth curve**, w.r.t. a connection in the tangent bundle,
is a smooth vector field along the curve with derivative zero:
$V \in \mathfrak{X}(\gamma)$, $D_t V = 0$.
The velocity of a geodesic is a vector field parallel along the curve.
**Parallel transport** $V$ (平行移动/平行输运) of a tangent vector along a (piecewise) smooth curve
is a parallel vector field along the curve that is compatible with the vector:
$\gamma: I \mapsto M$, $v \in T_{\gamma(t_0)} M$,
$V \in \mathfrak{X}(\gamma)$, $V(t_0) = v$, $D_t V = 0$.
Parallel transport exists and is unique for every smooth curve and every tangent vector on the curve.
**Parallel transport map** $P^{\gamma}_{t_0 t_1}$ between two tangent spaces on a smooth curve
is the map that takes every tangent vector at the initial point
to the tangent vector at the final point in parallel transport:
$P^{\gamma}_{t_0 t_1}: T_{\gamma(t_0)} M \mapsto T_{\gamma(t_1)} M$,
$\forall v \in T_{\gamma(t_0)} M$, $P^{\gamma}_{t_0 t_1}(v) = V(t_1)$.
The parallel transport map between every pair of tangent spaces on a smooth curve
is a linear isomorphism.
The parallel transport map is the means by which a connection "connects" nearby tangent spaces.
**Parallel frame** $(E_i)_{i=1}^n$ along a (piecewise) smooth curve in a smooth n-manifold
is the n-tuple of parallel transports of a basis along the curve:
$\text{Span}(b_i)_{i=1}^n = T_{\gamma(t_0)} M$, $\forall i \in n$, $E_i(t_0) = b_i$, $D_t E_i = 0$.
Every parallel frame along a curve is a frame along the curve,
i.e. at every point on the curve it is a basis of the tangent space.
Parallel transport determines covariant differentiation:
$\forall \gamma \in C^\infty(I, M)$, $\forall V \in \mathfrak{X}(\gamma)$, $\forall t_0 \in I$,
$D_t V(t_0) = \lim_{t_1 \to t_0} \frac{P^{\gamma}_{t_1 t_0} V(t_1) - V(t_0)}{t_1 - t_0}$.
Parallel transport determines the connection:
$\forall X, Y \in \mathfrak{X}(M)$, $\forall p \in M$,
$\forall \gamma \in \{l \in C^\infty(I, M) : l(0) = p, l'(0) = X_p \}$,
$\nabla_X Y|_p = \lim_{h \to 0} \frac{P^{\gamma}_{h 0} Y_{\gamma(h)} - Y_p}{h}$.
**Parallel vector field** on a smooth manifold, w.r.t. a connection in the tangent bundle,
is a smooth vector field that is parallel along every smooth curve in the manifold.
A smooth vector field is parallel if and only if
its total covariant derivative is zero: $\nabla V = 0$.

**Line integral** $\int_J \omega$ of a smooth covector field over a compact interval
is the ordinary integral of the standard coordinate representation of the field over the interval:
$J = [a, b]$, $\omega \in \mathfrak{X}^∗ (J)$, $\omega_t = \hat \omega(t) d t$, then
$\int_J \omega = \int_a^b \hat \omega(t) dt$.
**Line integral** $\int_\gamma \omega$ of a smooth covector field over a smooth curve segment
is the integral of the pullback of the field by the curve:
$\forall \omega \in \mathfrak{X}^∗ (M)$, $\forall \gamma \in C^\infty(J, M)$,
$\int_\gamma \omega = \int_J \gamma^∗ \omega$.
Rewinding the definitions, the line integral equals the ordinary integral of
the action of the covector field on curve velocity over the parameter interval:
$\int_\gamma \omega = \int_a^b \omega_{\gamma(t)}(\gamma'(t))~dt$.

The line integral of the pullback of a smooth covector field over a piecewise smooth curve segment equals the line integral of the field over the composite curve: $\forall F \in C^\infty(M, N)$, $\forall \eta \in \mathfrak{X}^∗ (N)$, $\forall \gamma \in C^\infty(J, M)$, $\int_\gamma F^∗ \eta = \int_{F \circ \gamma} \eta$.

**Reparameterization** $\tilde{\gamma}$ of a piecewise smooth curve segment $\gamma$
by a strictly monotonic smooth function $\phi: \tilde{J} \mapsto J$
is the composition of the curve with the bijection: $\tilde{\gamma} = \gamma \circ \phi$.
**Forward reparameterization** is a reparameterization by an increasing function.
**Backward reparameterization** is a reparameterization by a decreasing function.
The line integral of a smooth covector field over a piecewise smooth curve segment
is invariant under forward reparameterization, and flips sign under backward reparameterization.
*Fundamental Theorem for Line Integrals*:
The line integral of the differential of a smooth real-valued function
over a piecewise smooth curve segment equals the difference of function values
at the ends of the curve: $\int_\gamma d f = f(\gamma(b)) - f(\gamma(a))$.

Integral of real-valued functions on oriented smooth manifolds
cannot be defined independent of coordinates;
however, integral can be defined intrinsically for differential forms.
**Domain of integration** in a Euclidean space is a bounded subset whose boundary has measure zero.
**Integral** $\int_D \omega$ of an n-form $\omega$
on the closure of a domain of integration in the n-dimensional Euclidean space
is the integral of the standard coordinate representation of the n-form over the domain:
$\omega = f (\wedge_{i=1}^n dx^i)$, then $\int_D \omega = \int_D f d V$,
where $d V = \prod_{i=1}^n dx^i$.
Any compact subset $K$ of an open subset $U$ of a Euclidean space or a closed upper half-space
is included in an open domain of integration $D$ whose closure is also a subset of the open set:
$K \subset D \subset \bar{D} \subset U$.
**Integral** $\int_U \omega$ of a compactly supported n-form
on an open subset of a Euclidean space or a closed upper half-space
is the integral of the n-form on any domain of integration containing its support:
$\text{supp}~\omega = K$, $K \subset D \subset \bar{D} \subset U$, then
$\int_U \omega = \int_D \omega$.
The integral of the pullback of a compactly supported n-form
by an orientation-preserving diffeomorphism between open subsets of
a Euclidean space or its closed upper half-space
equals the integral of the n-form over the codomain: $\int_U F^∗ \omega = \int_{F(U)} \omega$;
the integral flips sign if the diffeomorphism is orientation-reversing.
**Integral** $\int_U \omega$ of an n-form on a compact subset
of a positively-oriented smooth coordinate domain of an oriented smooth n-manifold
is the integral of the pullback of the n-form by the inverse of the chart:
$\int_U \omega = \int_{\phi(U)} (\phi^{-1})^∗ \omega$;
the integral flips sign if the chart is negatively-oriented.
**Integral** $\int_M \omega$ of a compactly-supported n-form on an oriented smooth n-manifold
is the sum of integrals of n-forms $\psi_i \omega$,
where $\{\psi_i\}_{i=1}^m$ is any smooth partition of unity subordinate to a finite open cover
of the support of the n-form by positively or negatively oriented smooth charts:
$\int_M \omega = \sum_{i=1}^m \int_{U_i} \psi_i \omega$.
*Integration Over Piecewise Parameterizations*:
Integral of a compactly-supported n-form on an oriented smooth n-manifold
equals the sum of integrals of the n-form on a finite partition of its support
such that there are positively-oriented smooth charts
from their interior onto open domains of integration in the Euclidean n-space:
$\text{supp}~\omega = K = \overline{\sqcup_{i=1}^m U_i}$, $\phi_i: U_i \cong D_i$,
then $\int_M \omega = \sum_{i=1}^m \int_{D_i} (\phi_i^{-1})^∗ \omega$.
Integration over piecewise parameterizations also works for
boundary integrals of (n-1)-forms on any compact, oriented smooth n-manifold with corners,
and integrals of densities on any compact smooth n-manifold.
The integral map on compactly-supported n-forms on oriented smooth n-manifolds
is a linear functional that is positive for positively-oriented orientation forms,
is invariant under orientation-preserving diffeomorphisms,
and flips sign upon orientation reversal.

**Riemannian volume form** $\omega_g$ or $d V_g$ of an oriented Riemannian n-manifold $(M, g)$
is the unique n-form on the manifold satisfying any of the following equivalent properties:
(1) it equals the wedge product of any oriented orthonormal coframe, $\omega_g = \wedge_i ε^i$;
(2) it maps any oriented orthonormal frame to one, $\omega_g(e_i) = 1$;
(3) it equals the wedge product of any oriented coordinate coframe multiplied by
the square root of the determinant of the matrix representation of the Riemannian metric,
$\omega_g = \sqrt{\det g_{ij}} (\wedge_i dx^i)$.
The notation $d V_g$ for a Riemannian volume form or a Riemannian density
is just a convention, which does not mean it is the exterior derivative of an (n-1)-form.
The boundary of an oriented Riemannian manifold is orientable if and only if
there exists a global unit normal vector field on the boundary.
The Riemannian volume form of the boundary of an oriented Riemannian manifold,
given a global unit normal vector field,
is the interior multiplication of the Riemannian volume form of the manifold by the vector field:
$\omega_{\iota^∗ g} = (N \lrcorner \omega_g)|_{\partial M}$.
**Integral** $\int_M f \omega_g$ of a compactly-supported continuous real-valued function
over an oriented Riemannian manifold
is the integral of the compactly-supported n-form $f \omega_g$ on the manifold.
**Volume** $\text{Vol}(M)$ of a compact, oriented Riemannian manifold
is the integral of the Riemannian volume form on the manifold: $\text{Vol}(M) = \int_M d V_g$.

**Integral** $\int_S \omega$ of a k-form on an oriented smooth n-manifold
over an oriented smooth k-submanifold where the restriction of the form is compactly supported
is the integral of the pullback of the k-form by the inclusion map of the k-submanifold:
$\int_S \omega = \int_S \iota_S^∗ \omega$.
*Stokes’s Theorem*: $\int_M d \omega = \int_{\partial M} \omega$.

**Surface integral** $\int_S \langle X, N \rangle_g dA$
of a smooth vector field $X$ over a compact oriented 2-dimensional smooth submanifold $S$
with boundary in an oriented Riemannian 3-manifold.
*Stokes’s Theorem for Surface Integrals*:
$\int_S \langle \text{curl}~X, N \rangle_g dA = \int_{\partial S} \langle X, T \rangle_g ds$.

**Density** $\mu: V^n \mapsto \mathbb{R}$ on an n-dimensional vector space
is an n-variate real-valued function such that
its action on linearly transformed vectors equls its action on the original vectors,
multiplied by the absolute value of the determinant of the linear transformation:
$\forall T \in \mathcal{L}(V, V)$, $\mu(T v_i)_{i=1}^n = |\det T| \mu(v_i)_{i=1}^n$.
A density is not a tensor, because it is not linear over the real numbers in any of its arguments.
**Density space** $\mathcal{D}(V)$ on an n-dimensional vector space
is the vector space consisting of the set of all densities on the space,
and pointwise addition and scalar multiplication.
The density space on an n-dimensional vector space is the 1-dimensional vector space
consisting of the absolute value map and the negative value map of the n-covectors
on the underlying space:
$\mathcal{D}(V) = \{|\omega|, -|\omega| : \omega \in \Lambda^n(V^∗)\}$.
**Positive density** on an n-dimensional vector space
is one whose values are positive on a basis: $\mu = |\omega|$.
**Negative density** is defined analogously: $\mu = -|\omega|$.

**Density bundle** $\mathcal{D} M$ of a smooth manifold
is the disjoint union of density spaces on all tangent spaces of the manifold,
endowed with the natural projection map taking each point-indexed density to its point of tangent.
The density bundle of a smooth manifold is a smooth line bundle over the manifold.
**Density** $\mu$ on a smooth manifold is a section of the density bundle of the manifold.
**Positive density** on a smooth manifold is one
whose values are positive densities on all the tangent spaces.
Any nonvanishing n-form determines a positive density by taking pointwise absolute value:
$|\omega|_p = |\omega_p|$.
Any density can be written as a positive density multiplied by a real-valued function:
$\mu = f |\omega|$.
Every smooth manifold admits a smooth positive density.
**Pullback** $F^∗ \mu$ of a density on $N$ by a smooth map $F \in C^\infty(M, N)$
is the density on $M$ whose value at each point equals the pullback of the density at that point:
$\forall v_i \in T_p M$, $(F^∗ \mu)_p (v_i)_{i=1}^n = \mu_{F(p)}(d F_p (v_i))_{i=1}^n$.
The pullback of a smooth density by a smooth map is a smooth density.

**Integral** $\int_D \mu$ of a density
on (the closure of) a domain of integration in the Euclidean n-space
is the integral of the standard coordinate representation of the density over the domain:
$\mu = f |\wedge_{i=1}^n dx^i|$, then $\int_D \mu = \int_D f d V$ where $d V = \prod_{i=1}^n dx^i$.
Analogous to integral of n-forms, we have:
**integral** of a compacly supported density on an open subset, $\int_U \mu = \int_D \mu$,
which is diffeomorphism-invariant, $\int_U F^∗ \mu = \int_{F(U)} \mu$;
**integral** on a compact subset of a smooth coordinate domain of a smooth manifold,
$\int_U \mu = \int_{\phi(U)} (\phi^{-1})^∗ \mu$;
**integral** of a compactly-supported density on a smooth manifold via a smooth partition of unity,
$\int_M \mu = \sum_i \int_M \psi_i \mu$.
The integral map on compactly-supported densities on smooth manifolds
is a linear functional that is positive for positive densities
and is invariant under diffeomorphisms.

**Riemannian density** $\mu_g$ or $d V_g$ on a Riemannian manifold $(M, g)$
is the unique smooth positive density that maps any orthonormal frame to one: $\mu_g(e_i) = 1$.
**Integral** $\int_M f \mu_g$ of a compactly-supported continuous real-valued function
over a Riemannian manifold is the integral of the density $f \mu_g$ on the manifold.
For an oriented Riemannian manifold, its Riemannian density equals
the absolute value map of its Riemannian volume form: $\mu_g = |\omega_g|$,
and thus the integrals of a function as a density and an n-form are the same:
$\int_M f \mu_g = \int_M f \omega_g$.
*Divergence Theorem*:
The integral of the divergence of a compactly-supported smooth vector field
on a Riemannian manifold with boundary
equals the integral of the inner product of the vector field and
the outward-pointing unit normal vector field along the manifold boundary:
$\int_M (\text{div}~X) \mu_g = \int_{\partial M} \langle X, N \rangle_g \mu_{\tilde{g}}$,
where $\tilde{g}$ is the induced Riemannian metric on the manifold boundary.

Measure on smooth/Riemannian manifolds. Directional statistics deals with observations on n-spheres [@Brigant2019]. Sampling on manifolds [@Soize2016]. Measures on a smooth manifold are preserved in piecewise parameterizations, despite violating the topology. A unit n-volume $I^n$ should suffice for sampling on a connected compact Riemannian n-manifold, e.g. sampling on a sphere via geographic coordinates.