Riemannian Metrics Length Distance and Volume — Examples

Given: $g={\sum }_{i}d{x}^i\otimes d{x}^i$, $v={\sum }_i{v}^i{\mathrm{\partial }}_i$, and $\alpha ={\sum }_i{\alpha }_{i}d{x}^i$.

Coordinate criterion for a riemannian metric: A tensor $g={\sum }_{i,j}{g}_{ij}d{x}^i\otimes d{x}^j$ is Riemannian exactly when its coordinate matrix $G=(g_{ij})$ has smooth entries and is symmetric positive definite. Under $J=\mathrm{\partial }x\mathrm{/}\mathrm{\partial }y$ it transforms by $G_y=J^T{G}_{x}J$.

The musical maps are smooth inverse bundle isomorphisms: $\mathrm{♭}:TM\to T^{\ast }M$ and $\mathrm{♯}:T^{\ast }M\to TM$ are smooth inverse bundle isomorphisms.

The gradient is characterized by inner products: The gradient is the unique smooth vector field $Y$ satisfying $g(Y,X)=Xf$ for every smooth vector field $X$.

Musical isomorphisms: The musical maps are defined by $v^{\mathrm{♭}}=g(v,\cdot )$ and by $g({\alpha }^{\mathrm{♯}},w)=\alpha (w)$ for every tangent vector $w$.

Given: $S^n=\{p\in {\mathbb{R}}^{n+1}:\mathrm{\parallel }p\mathrm{\parallel }=1\}$ and its usual smooth structure; $i$ is inclusion.

Pullback of a riemannian metric is riemannian exactly for immersions: $F^{\ast }h$ is Riemannian if and only if $F$ is an immersion. In general it is positive semidefinite, with radical $\ker d{F}_p$ at $p$.

Coordinate criterion for a riemannian metric: A tensor $g={\sum }_{i,j}{g}_{ij}d{x}^i\otimes d{x}^j$ is Riemannian exactly when its coordinate matrix $G=(g_{ij})$ has smooth entries and is symmetric positive definite. Under $J=\mathrm{\partial }x\mathrm{/}\mathrm{\partial }y$ it transforms by $G_y=J^T{G}_{x}J$.

Given: Finite-dimensional smooth Riemannian manifolds $(M,g),(N,h)$, with their product smooth structure.

Coordinate criterion for a riemannian metric: A tensor $g={\sum }_{i,j}{g}_{ij}d{x}^i\otimes d{x}^j$ is Riemannian exactly when its coordinate matrix $G=(g_{ij})$ has smooth entries and is symmetric positive definite. Under $J=\mathrm{\partial }x\mathrm{/}\mathrm{\partial }y$ it transforms by $G_y=J^T{G}_{x}J$.

Products of smooth manifolds have a canonical product smooth structure: Let $(M,\mathcal{S})$ and $(N,\mathcal{T})$ be smooth manifolds of dimensions $m$ and $n$. Then $M\times N$ with the product topology is a topological $(m+n)$-manifold. If $\mathcal{A}$ and $\mathcal{B}$ are smooth atlases with $[\mathcal{A}]=\mathcal{S}$ and $[\mathcal{B}]=\mathcal{T}$, then the set of product charts $\mathcal{A}\times \mathcal{B}:=\left\{\right(V\times W,\phi \times \psi ):(V,\phi )\in \mathcal{A},(W,\psi )\in \mathcal{B}\}$ is a smooth atlas on $M\times N$, and the maximal atlas it generates is independent of the presenting atlases: it depends only on $\mathcal{S}$ and $\mathcal{T}$. This maximal atlas is the product smooth structure of $M\times N$.

Given: $u,f\in C^{\mathrm{\infty }}({\mathbb{R}}^2)$.

Conformal equivalence of riemannian metrics: Two Riemannian metrics are conformally equivalent if $\tilde{g}=e^{2u}g$ for a smooth real function $u$ on $M$. The positive smooth factor preserves the metric condition of def-riemannian-metric-and-riemannian-manifold. Equivalently $\tilde{g}=fg$ for smooth $f>0$, since $u=\frac{1}{2}\log f$. Reflexivity uses $u=0$, reversal uses $-u$, and composing rescalings adds their functions.

Riemannian gradient: For a smooth real function $f$, its Riemannian gradient is ${\mathrm{grad}}_{g}f=(df{)}^{\mathrm{♯}}$. prop-exterior-derivative-of-a-function-is-its-differential identifies $df(X)=Xf$. The smooth bundle isomorphism in thm-the-musical-maps-are-smooth-inverse-bundle-isomorphisms therefore makes the gradient a smooth vector field. In coordinates $({\mathrm{grad}}_{g}f{)}^i={\sum }_j{g}^{ij}{\mathrm{\partial }}_{j}f$. Constants, and all functions in dimension zero, have zero gradient.

Riemannian volume density: The Riemannian volume density is ${\mu }_g=\sqrt{\det G_x}\mathrm{\mid }d{x}^1\cdots d{x}^n\mathrm{\mid }$ in coordinates. The matrix is that of prop-coordinate-criterion-for-a-riemannian-metric, so its determinant is positive and smooth. The density frames and their absolute-Jacobian law are def-density-bundle-and-smooth-density. In dimension zero take the empty determinant to be one, giving weight one at every point, independently of orientation. The compatibility of these local formulas is proved in lem-the-riemannian-volume-density-is-coordinate-independent.

Given: The upper half-space with its Euclidean open-subset smooth structure.

Coordinate criterion for a riemannian metric: A tensor $g={\sum }_{i,j}{g}_{ij}d{x}^i\otimes d{x}^j$ is Riemannian exactly when its coordinate matrix $G=(g_{ij})$ has smooth entries and is symmetric positive definite. Under $J=\mathrm{\partial }x\mathrm{/}\mathrm{\partial }y$ it transforms by $G_y=J^T{G}_{x}J$.

Riemannian volume density: The Riemannian volume density is ${\mu }_g=\sqrt{\det G_x}\mathrm{\mid }d{x}^1\cdots d{x}^n\mathrm{\mid }$ in coordinates. The matrix is that of prop-coordinate-criterion-for-a-riemannian-metric, so its determinant is positive and smooth. The density frames and their absolute-Jacobian law are def-density-bundle-and-smooth-density. In dimension zero take the empty determinant to be one, giving weight one at every point, independently of orientation. The compatibility of these local formulas is proved in lem-the-riemannian-volume-density-is-coordinate-independent.

Given: The integer translation action on ${\mathbb{R}}^n$ and quotient map $q$.

Coordinate criterion for a riemannian metric: A tensor $g={\sum }_{i,j}{g}_{ij}d{x}^i\otimes d{x}^j$ is Riemannian exactly when its coordinate matrix $G=(g_{ij})$ has smooth entries and is symmetric positive definite. Under $J=\mathrm{\partial }x\mathrm{/}\mathrm{\partial }y$ it transforms by $G_y=J^T{G}_{x}J$.

Given: Real angles $a,b$, and the circle parametrization $e^{it}=(\cos t,\sin t)$.

Riemannian speed and length: The Riemannian speed on a $C^1$ piece is $\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}_g=\sqrt{g_{\gamma (t)}(\dot{\gamma }(t),\dot{\gamma }(t))}$. Its length is $L_g(\gamma )={\sum }_j{\int }_{t_{j-1}}^{t_j}\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}_{g}dt$. The curve convention is def-piecewise-c-one-curve-on-a-manifold and the norm is def-pointwise-norm-and-angle-from-a-riemannian-metric. Each integrand is continuous on its closed piece with the one-sided endpoint derivative, hence Riemann integrable and nonnegative. Values chosen at the finitely many corners do not change its integral. For a singleton interval the empty sum is zero; a constant curve also has zero length. Partition independence is established next.

Riemannian distance on a connected manifold: On a connected Riemannian manifold define $d_g(p,q)=\inf \{L_g(\gamma ):\gamma \text{ is piecewise }{C}^1\text{ from }p\text{ to }q\}$. Lengths are those of def-riemannian-speed-and-length. For each pair $p,q$, lem-any-two-points-in-a-connected-smooth-manifold-can-be-joined-by-a-piecewise-c-one-curve supplies a curve, so the set of lengths is nonempty, contains a finite real number and is bounded below by zero. Applying the least-upper-bound property cor-cauchy-reals-lub-complete to the negatives gives a finite nonnegative infimum. On the empty connected manifold this defines the empty distance function; there are no pairs to evaluate. No minimizing curve is part of this definition.

Newton–Leibniz needs only continuity on $[a,b]$, differentiability on $(a,b)$, and a Riemann-integrable extension of the interior derivative: Let $a<b$. Suppose $G:[a,b]\to \mathbb{R}$ is continuous on $[a,b]$ and differentiable on $(a,b)$. If $f:[a,b]\to \mathbb{R}$ is Riemann integrable and $f(x)=G^{\mathrm{\prime }}(x)(a<x<b),$ then ${\int }_a^{b}f=G(b)-G(a)\mathrm{.}$ No derivative of $G$ at either endpoint is assumed, and the two endpoint values assigned to the integrable extension $f$ do not enter the conclusion.

Given: The two disjoint Euclidean lines with their disjoint-union smooth structure.

Extended riemannian distance on a disconnected manifold: The extended Riemannian distance on arbitrary $M$ is the componentwise Riemannian distance when two points are in the same component, and $+\mathrm{\infty }$ otherwise. Within each component use thm-riemannian-distance-is-a-metric. Components are open, since small coordinate balls are connected. A continuous curve cannot meet two components because its connected interval image is connected, so the cross-component curve family is empty, with $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. This is an extended metric: if two endpoints are in different components, any third point is in a different component from at least one of them, so the triangle inequality has infinite right side. It is a finite metric precisely when there are no distinct components. Empty and singleton manifolds retain their unique distances.

Newton–Leibniz needs only continuity on $[a,b]$, differentiability on $(a,b)$, and a Riemann-integrable extension of the interior derivative: Let $a<b$. Suppose $G:[a,b]\to \mathbb{R}$ is continuous on $[a,b]$ and differentiable on $(a,b)$. If $f:[a,b]\to \mathbb{R}$ is Riemann integrable and $f(x)=G^{\mathrm{\prime }}(x)(a<x<b),$ then ${\int }_a^{b}f=G(b)-G(a)\mathrm{.}$ No derivative of $G$ at either endpoint is assumed, and the two endpoint values assigned to the integrable extension $f$ do not enter the conclusion.

Given: $x=r\cos \theta$, $y=r\sin \theta$, with $r>0$ and $\theta$ in an open interval of length less than $2\pi$.

Riemannian volume density: The Riemannian volume density is ${\mu }_g=\sqrt{\det G_x}\mathrm{\mid }d{x}^1\cdots d{x}^n\mathrm{\mid }$ in coordinates. The matrix is that of prop-coordinate-criterion-for-a-riemannian-metric, so its determinant is positive and smooth. The density frames and their absolute-Jacobian law are def-density-bundle-and-smooth-density. In dimension zero take the empty determinant to be one, giving weight one at every point, independently of orientation. The compatibility of these local formulas is proved in lem-the-riemannian-volume-density-is-coordinate-independent.

The riemannian volume density is coordinate independent: The local Riemannian volume densities glue to a positive smooth density independent of coordinates.

Given: The orthonormal coframe $(dx,dy,dz)$ with volume $\nu =dx\wedge dy\wedge dz$.

Hodge star is a smooth bundle isomorphism: The Hodge star exists uniquely and is a smooth bundle isomorphism in every degree $0\le k\le n$.

Hodge star squared sign: On real $k$-forms, ${\ast }^2=(-1{)}^{k(n-k)}\mathrm{id}$.

Given: A polar chart with $r>0$, metric matrix $\mathrm{diag}(1,r^2)$, and a smooth vector field $X$.

Coordinate formula for riemannian divergence: In coordinates, ${\mathrm{div}}_{g}X=(\det G{)}^{-1\mathrm{/}2}{\sum }_{i=1}^n{\mathrm{\partial }}_i((\det G{)}^{1\mathrm{/}2}{X}^i)$.

Given: $F:M\to \mathbb{R}$ is constant, $M$ is a smooth manifold of dimension $n>0$, and the target metric is $d{y}^2$.

Pullback of a riemannian metric as a tensor: For smooth $F:M\to N$ and a Riemannian metric $h$ on $N$, its pullback tensor is $(F^{\ast }h{)}_p(v,w)=h_{F(p)}(d{F}_{p}v,d{F}_{p}w)$. This is def-pullback-of-a-covariant-tensor-field for the tensor in def-riemannian-metric-and-riemannian-manifold. It is always symmetric and positive semidefinite; the name does not assert positive definiteness. Smoothness and the precise immersion criterion are established next.

Pullback of a riemannian metric is riemannian exactly for immersions: $F^{\ast }h$ is Riemannian if and only if $F$ is an immersion. In general it is positive semidefinite, with radical $\ker d{F}_p$ at $p$.

Smooth manifolds and their smooth charts: A smooth $n$-manifold is a pair $(M,\mathcal{S})$ in which $M$ is a topological $n$-manifold (def-topological-manifold-without-boundary) and $\mathcal{S}$ is a smooth structure on $M$: a maximal smooth atlas (thm-each-smooth-atlas-is-contained-in-a-unique-maximal-smooth-atlas). Because thm-each-smooth-atlas-is-contained-in-a-unique-maximal-smooth-atlas sends every smooth atlas to the unique maximal atlas containing it, a smooth manifold is equivalently specified by a topological manifold $M$ together with any one smooth atlas $\mathcal{A}$, the structure being the generated $[\mathcal{A}]$. A chart $(U,\phi )\in \mathcal{S}$ is called a smooth chart (or a chart of the smooth structure); its domain is a coordinate domain and its coordinate functions are smooth coordinates on $U$. When the structure is clear from context, the manifold itself is written $M$ in place of $(M,\mathcal{S})$.

Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces: Let $n\in \mathbb{N}$. For $n\ge 1$, put ${\mathbb{R}}^n:={\prod }_{k<n}\mathbb{R}$ with its usual topology; for $n=0$ put ${\mathbb{R}}^0:=\{0\}$, the one-point space. A topological $n$-manifold without boundary (or briefly an $n$-manifold) is a topological space $M$ satisfying: 1. $M$ is Hausdorff (def-hausdorff-space); 2. $M$ is second countable (def-second-countable-space); 3. $M$ is locally Euclidean of dimension $n$: every $p\in M$ has an open neighbourhood $U\subseteq M$ homeomorphic to an open subset of ${\mathbb{R}}^n$ (def-homeomorphism-and-open-maps). The empty space satisfies all three conditions vacuously, so $M=\mathrm{\varnothing }$ is an $n$-manifold for every $n$; this degenerate instance is kept, and statements about nonempty manifolds name the hypothesis. In dimension zero, condition 3 forces the one-point neighbourhoods of points to be open singletons, so a $0$-manifold is exactly a discrete second-countable space with at most countably many points.