Calculus can be defined on smooth manifolds.

## Dynamical System

### Vector Field

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.

### Flow

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 $p$: $V|_U = \partial/\partial x^1$, $S \cap U = (x^1)^{-1}(0)$.

### First-order Cauchy Problems

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

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 PDEs

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

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

### Covector Field

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

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.

### Exterior Derivative

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.

### Derived Differential Operators

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

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

### Covariant Derivative to a Vector Field

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 $(\nabla (\nabla u))^\sharp = (\nabla (d u))^\sharp = \nabla ((d u)^\sharp) = \nabla (\text{grad}~u)$.

### Covariant Derivative along a Curve

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: $\nabla V = 0$.

## Integration

### Integral of 1-Form on Curve Segment

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 n-Form on Oriented n-Manifold

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 of k-Form on Oriented k-Submanifold

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: $\int_S \langle \text{curl}~X, N \rangle_g dA = \int_{\partial S} \langle X, T \rangle_g ds$.

### Integral of Density on Manifold

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.