This article is about a concept from differential geometry. For the algebraic concept, see
Zariski–Riemann space.
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold M equipped with an inner product $g\_p$ on the tangent space $T\_pM$ at each point $p$
that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then
$p\; \backslash mapsto\; g\_p(X(p),Y(p))$ is a smooth function.
The family $g\_p$ of inner products is called a Riemannian metric (tensor).
These terms are named after the German mathematician Bernhard Riemann.
The study of Riemannian manifolds comprises the subject called Riemannian geometry.
A Riemannian metric (tensor) makes it possible to define various geometric notions on a Riemannian manifold, such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields.
Introduction
In 1828, Carl Friedrich Gauss proved his Theorema Egregium (remarkable theorem in Latin), establishing an important property of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space. See differential geometry of surfaces. Bernhard Riemann extended Gauss's theory to higher dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. Albert Einstein used the theory of Riemannian manifolds to develop his general theory of relativity. In particular, his equations for gravitation are restrictions on the curvature of space.
Overview
The tangent bundle of a smooth manifold M assigns to each fixed point of M a vector space called the tangent space, and each tangent space can be equipped with an inner product. If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold. For example, a smooth curve α(t): [0, 1] → M has tangent vector α′(t_{0}) in the tangent space TM(α(t_{0})) at any point t_{0} ∈ (0, 1), and each such vector has length ‖α′(t_{0})‖, where ‖·‖ denotes the norm induced by the inner product on TM(α(t_{0})). The integral of these lengths gives the length of the curve α:
- $L(\backslash alpha)\; =\; \backslash int\_0^1\{\backslash |\backslash alpha\text{'}(t)\backslash |\backslash ,\; \backslash mathrm\{d\}t\}.$
Smoothness of α(t) for t in [0, 1] guarantees that the integral L(α) exists and the length of this curve is defined.
In many instances, in order to pass from a linear-algebraic concept to a differential-geometric one, the smoothness requirement is very important.
Every smooth submanifold of R^{n} has an induced Riemannian metric g: the inner product on each tangent space is the restriction of the inner product on R^{n}. In fact, as follows from the Nash embedding theorem, all Riemannian manifolds can be realized this way.
In particular one could define Riemannian manifold as a metric space which is isometric to a smooth submanifold of R^{n} with the induced intrinsic metric, where isometry here is meant in the sense of preserving the length of curves. This definition might theoretically not be flexible enough, but it is quite useful to build the first geometric intuitions in Riemannian geometry.
Riemannian manifolds as metric spaces
Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of the positive-definite quadratic forms on the tangent bundle. Then one has to work to show that it can be turned to a metric space:
If γ: [a, b] → M is a continuously differentiable curve in the Riemannian manifold M, then we define its length L(γ) in analogy with the example above by
- $L(\backslash gamma)\; =\; \backslash int\_a^b\; \backslash |\backslash gamma\text{'}(t)\backslash |\backslash ,\; \backslash mathrm\{d\}t.$
With this definition of length, every connected Riemannian manifold M becomes a metric space (and even a length metric space) in a natural fashion: the distance d(x, y) between the points x and y of M is defined as
- d(x,y) = inf{ L(γ) : γ is a continuously differentiable curve joining x and y}.
Even though Riemannian manifolds are usually "curved," there is still a notion of "straight line" on them: the geodesics. These are curves which locally join their points along shortest paths.
Assuming the manifold is compact, any two points x and y can be connected with a geodesic whose length is d(x,y). Without compactness, this need not be true. For example, in the punctured plane R^{2} \ {0}, the distance between the points (−1, 0) and (1, 0) is 2, but there is no geodesic realizing this distance.
Properties
In Riemannian manifolds, the notions of geodesic completeness, topological completeness and metric completeness are the same: that each implies the other is the content of the Hopf–Rinow theorem.
Riemannian metrics
Let M be a differentiable manifold of dimension n. A Riemannian metric on M is a family of (positive definite) inner products
- $g\_p\; \backslash colon\; T\_pM\backslash times\; T\_pM\backslash longrightarrow\; \backslash mathbf\; R,\backslash qquad\; p\backslash in\; M$
such that, for all differentiable vector fields X,Y on M,
- $p\backslash mapsto\; g\_p(X(p),\; Y(p))$
defines a smooth function M → R.
In other words, a Riemannian metric g is a symmetric (0,2)-tensor that is positive definite (i.e. g(X, X) > 0 for all tangent vectors X ≠ 0).
In a system of local coordinates on the manifold M given by n real-valued functions x^{1},x^{2}, …, x^{n}, the vector fields
- $\backslash left\backslash \{\backslash frac\{\backslash partial\}\{\backslash partial\; x^1\},\backslash dotsc,\; \backslash frac\{\backslash partial\}\{\backslash partial\; x^n\}\backslash right\backslash \}$
give a basis of tangent vectors at each point of M. Relative to this coordinate system, the components of the metric tensor are, at each point p,
- $g\_\{ij\}(p):=g\_p\backslash Biggl(\backslash left(\backslash frac\{\backslash partial\; \}\{\backslash partial\; x^i\}\backslash right)\_p,\backslash left(\backslash frac\{\backslash partial\; \}\{\backslash partial\; x^j\}\backslash right)\_p\backslash Biggr).$
Equivalently, the metric tensor can be written in terms of the dual basis {dx^{1}, …, dx^{n}} of the cotangent bundle as
- $g=\backslash sum\_\{i,j\}g\_\{ij\}\backslash mathrm\; d\; x^i\backslash otimes\; \backslash mathrm\; d\; x^j.$
Endowed with this metric, the differentiable manifold (M, g) is a Riemannian manifold.
Examples
- With $\backslash frac\{\backslash partial\; \}\{\backslash partial\; x^i\}$ identified with e_{i} = (0, …, 1, …, 0), the standard metric over an open subset U ⊂ R^{n} is defined by
- $g^\{\backslash mathrm\{can\}\}\_p\; \backslash colon\; T\_pU\backslash times\; T\_pU\backslash longrightarrow\; \backslash mathbf\; R,\backslash qquad\; \backslash left(\backslash sum\_ia\_i\backslash frac\{\backslash partial\}\{\backslash partial\; x^i\},\backslash sum\_jb\_j\backslash frac\{\backslash partial\}\{\backslash partial\; x^j\}\backslash right)\backslash longmapsto\; \backslash sum\_i\; a\_ib\_i.$
- Then g is a Riemannian metric, and
- $g^\{\backslash mathrm\{can\}\}\_\{ij\}=\backslash langle\; e\_i,e\_j\backslash rangle\; =\; \backslash delta\_\{ij\}.$
- Equipped with this metric, R^{n} is called Euclidean space of dimension n and g_{ij}^{can} is called the (canonical) Euclidean metric.
- Let (M,g) be a Riemannian manifold and N ⊂ M be a submanifold of M. Then the restriction of g to vectors tangent along N defines a Riemannian metric over N.
- More generally, let f: M^{n}→N^{n+k} be an immersion. Then, if N has a Riemannian metric, f induces a Riemannian metric on M via pullback:
- $g^M\_p\; \backslash colon\; T\_pM\backslash times\; T\_pM\backslash longrightarrow\; \backslash mathbf\; R,$
- $(u,v)\backslash longmapsto\; g^M\_p(u,v):=g^N\_\{f(p)\}(T\_pf(u),\; T\_pf(v)).$
- This is then a metric; the positive definiteness follows on the injectivity of the differential of an immersion.
- Let (M, g^{M}) be a Riemannian manifold, h:M^{n+k}→N^{k} be a differentiable map and q∈N be a regular value of h (the differential dh(p) is surjective for all p∈h^{−1}(q)). Then h^{−1}(q)⊂M is a submanifold of M of dimension n. Thus h^{−1}(q) carries the Riemannian metric induced by inclusion.
- In particular, consider the following map :
- $h\backslash colon\; \backslash mathbf\; R^n\backslash longrightarrow\; \backslash mathbf\; R,\backslash qquad\; (x^1,\; \backslash dotsc,\; x^n)\backslash longmapsto\; \backslash sum\_\{i=1\}^n(x^i)^2-1.$
- Then, 0 is a regular value of h and
- $h^\{-1\}(0)=\; \backslash left\; \backslash \{x\backslash in\backslash mathbf\; R^n\backslash vert\; \backslash sum\_\{i=1\}^n(x^i)^2=1\; \backslash right\; \backslash \}=\; \backslash mathbf\{S\}^\{n-1\}$
- is the unit sphere S^{n − 1} ⊂ R^{n}. The metric induced from R^{n} on S^{n − 1} is called the canonical metric of S^{n − 1}.
- Let M_{1} and M_{2} be two Riemannian manifolds and consider the cartesian product M_{1} × M_{2} with the product structure. Furthermore, let π_{1}: M_{1} × M_{2} → M_{1} and π_{2}: M_{1} × M_{2} → M_{2} be the natural projections. For (p,q) ∈ M_{1} × M_{2}, a Riemannian metric on M_{1} × M_{2} can be introduced as follows :
- $g^\{M\_1\backslash times\; M\_2\}\_\{(p,q)\}\backslash colon\; T\_\{(p,q)\}(M\_1\backslash times\; M\_2)\backslash times\; T\_\{(p,q)\}(M\_1\backslash times\; M\_2)\; \backslash longrightarrow\; \backslash mathbf\; R,$
- $(u,v)\backslash longmapsto\; g^\{M\_1\}\_p(T\_\{(p,q)\}\backslash pi\_1(u),\; T\_\{(p,q)\}\backslash pi\_1(v))+g^\{M\_2\}\_q(T\_\{(p,q)\}\backslash pi\_2(u),\; T\_\{(p,q)\}\backslash pi\_2(v)).$
- The identification
- $T\_\{(p,q)\}(M\_1\backslash times\; M\_2)\; \backslash cong\; T\_pM\_1\backslash oplus\; T\_qM\_2$
- allows us to conclude that this defines a metric on the product space.
- The torus S^{1} × … × S^{1} = T^{n} possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from R^{2} on the circle S^{1} ⊂ R^{2} and then taking the product metric. The torus T^{n} endowed with this metric is called the flat torus.
- Let g_{0}, g_{1} be two metrics on M. Then,
- $\backslash tilde\; g:=\backslash lambda\; g\_0\; +\; (1-\backslash lambda)g\_1,\backslash qquad\; \backslash lambda\backslash in\; [0,1],$
- is also a metric on M.
The pullback metric
If f:M→N is a differentiable map and (N,g^{N}) a Riemannian manifold, then the pullback of g^{N} along f is a quadratic form on the tangent space of M. The pullback is the quadratic form f*g^{N} on TM defined for v, w ∈ T_{p}M by
- $(f^*g^N)(v,w)\; =\; g^N(df(v),df(w))\backslash ,.$
where df(v) is the pushforward of v by f.
The quadratic form f*g^{N} is in general only a semi definite form because df can have a kernel. If f is a diffeomorphism, or more generally an immersion, then it defines a Riemannian metric on M, the pullback metric. In particular, every embedded smooth submanifold inherits a metric from being embedded in a Riemannian manifold, and every covering space inherits a metric from covering a Riemannian manifold.
Existence of a metric
Every paracompact differentiable manifold admits a Riemannian metric. To prove this result, let M be a manifold and {(U_{α}, φ(U_{α}))|α ∈ I} a locally finite atlas of open subsets U of M and diffeomorphisms onto open subsets of R^{n}
- $\backslash phi\; \backslash colon\; U\_\backslash alpha\backslash to\; \backslash phi(U\_\backslash alpha)\backslash subseteq\backslash mathbf\{R\}^n.$
Let τ_{α} be a differentiable partition of unity subordinate to the given atlas. Then define the metric g on M by
- $g:=\backslash sum\_\backslash beta\backslash tau\_\backslash beta\backslash cdot\backslash tilde\{g\}\_\backslash beta,\backslash qquad\backslash text\{with\}\backslash qquad\backslash tilde\{g\}\_\backslash beta:=\backslash tilde\{\backslash phi\}\_\backslash beta^*g^\{\backslash mathrm\{can\}\}.$
where g^{can} is the Euclidean metric. This is readily seen to be a metric on M.
Isometries
Let (M, g^{M}) and (N, g^{N}) be two Riemannian manifolds, and f: M → N be a diffeomorphism. Then, f is called an isometry, if
- $g^M\; =\; f^*\; g^N\backslash ,,$
or pointwise
- $g^M\_p(u,v)\; =\; g^N\_\{f(p)\}(df(u),\; df(v))\backslash qquad\; \backslash forall\; p\backslash in\; M,\; \backslash forall\; u,v\backslash in\; T\_pM.$
Moreover, a differentiable mapping f: M → N is called a local isometry at p ∈ M if there is a neighbourhood U ⊂ M, p ∈ U, such that f: U → f(U) is a diffeomorphism satisfying the previous relation.
Riemannian manifolds as metric spaces
A connected Riemannian manifold carries the structure of a metric space whose distance function is the arclength of a minimizing geodesic.
Specifically, let (M,g) be a connected Riemannian manifold. Let c: [a,b] → M be a parametrized curve in M, which is differentiable with velocity vector c′. The length of c is defined as
- $L\_a^b(c):=\; \backslash int\_a^b\; \backslash sqrt\{g(c\text{'}(t),c\text{'}(t))\}\backslash ,\backslash mathrm\; d\; t\; =\; \backslash int\_a^b\backslash |c\text{'}(t)\backslash |\backslash ,\backslash mathrm\; d\; t.$
By change of variables, the arclength is independent of the chosen parametrization. In particular, a curve [a,b] → M can be parametrized by its arc length. A curve is parametrized by arclength if and only if $\backslash |c\text{'}(t)\backslash |=1$ for all $t\backslash in[a,b]$.
The distance function d : M×M → [0,∞) is defined by
- $d(p,q)\; =\; \backslash inf\; L(\backslash gamma)$
where the infimum extends over all differentiable curves γ beginning at p ∈ M and ending at q ∈ M.
This function d satisfies the properties of a distance function for a metric space. The only property which is not completely straightforward is to show that d(p,q) = 0 implies that p = q. For this property, one can use a normal coordinate system, which also allows one to show that the topology induced by d is the same as the original topology on M.
Diameter
The diameter of a Riemannian manifold M is defined by
- $\backslash mathrm\{diam\}(M):=\backslash sup\_\{p,q\backslash in\; M\}\; d(p,q)\backslash in\; \backslash mathbf\; R\_\{\backslash geq\; 0\}\backslash cup\backslash \{+\backslash infty\backslash \}.$
The diameter is invariant under global isometries. Furthermore, the Heine–Borel property holds for (finite-dimensional) Riemannian manifolds: M is compact if and only if it is complete and has finite diameter.
Geodesic completeness
A Riemannian manifold M is geodesically complete if for all p ∈ M, the exponential map $\backslash exp\_p$ is defined for all $v\backslash in\; T\_pM$, i.e. if any geodesic $\backslash gamma(t)$ starting from p is defined for all values of the parameter t ∈ R. The Hopf-Rinow theorem asserts that M is geodesically complete if and only if it is complete as a metric space.
If M is complete, then M is non-extendable in the sense that it is not isometric to an open proper submanifold of any other Riemannian manifold. The converse is not true, however: there exist non-extendable manifolds which are not complete.
See also
References
External links
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