Riemannian manifold

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In Riemannian geometry, a Riemannian manifold (M,g) (with Riemannian metric g) is a real differentiable manifold M in which each tangent space is equipped with an inner product g in a manner which varies smoothly from point to point. This allows one to define various notions such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields. In other words, a Riemannian manifold is a differentiable manifold in which the tangent space at each point is a finite-dimensional Hilbert space. The terms are named after German mathematician Bernhard Riemann.

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[edit] 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′(t0) in the tangent space TM(t0) at any point t0 ∈ (0, 1), and each such vector has length ||α′(t0)||, where ||·|| denotes the norm induced by the inner product on TM(t0). The integral of these lengths gives the length of the curve α:

L(\alpha) = \int_0^1{\|\alpha^{\prime}(t)\|\, \mathrm{d}t}.

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 Rn has an induced Riemannian metric g: the inner product on each tangent space is the restriction of the inner product on Rn. 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 Rn 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.

[edit] 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(\gamma) = \int_a^b \|\gamma'(t)\|\, \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 R2 \ {0}, the distance between the points (−1, 0) and (1, 0) is 2, but there is no geodesic realizing this distance.

[edit] 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.

[edit] Riemannian metrics

Let M be a second countable Hausdorff differentiable manifold of dimension n. A Riemannian metric on M is a family of inner products

g_p : T_pM\times T_pM\longrightarrow \mathbb R,\qquad p\in M

such that, for all differentiable vector fields X,Y\in\mathcal V(M), the application

M\longrightarrow \mathbb R,\qquad p\longmapsto g_p(X(p), Y(p))

is differentiable. Let \{\left(\frac{\partial }{\partial x_i}\right)_p\}_i be a basis of tangent vectors over p\in M. Then, the coefficients

g_{ij}(p):=\Big\langle\left(\frac{\partial }{\partial x_i}\right)_p,\left(\frac{\partial }{\partial x_j}\right)_p\Big\rangle_p

give rise to the metric tensor of rank 2

g:=\sum_{i,j}g_{ij}\mathrm d x_i\otimes \mathrm d x_j.

Endowed with this metric, the differentiable manifold (M,g) is a Riemannian manifold.

[edit] Examples

  • With \frac{\partial }{\partial x_i} identified with e_i=(0,\dots, 1,\dots, 0), the standard metric over an open subset U\subset\mathbb R^n is defined by
g^{\mathrm{can}}_p : T_pU\times T_pU\longrightarrow \mathbb R,\qquad \left(\sum_ia_i\frac{\partial}{\partial x_i},\sum_jb_j\frac{\partial}{\partial x_j}\right)\longmapsto \sum_i a_ib_i.
Then g is a Riemannian metric, and
g^{\mathrm{can}}_{ij}=\langle e_i,e_j\rangle = \delta_{ij}.
Equipped with this metric, Rn is called Euclidean space of dimension n and gijcan is called the Euclidean metric.
  • Let (M,g) be a Riemannian manifold and N\subset 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:Mn→Nn+k be an immersion. Then, if N has a Riemannian metric, f induces a Riemannian metric on M via pullback:
g^M_p : T_pM\times T_pM\longrightarrow \mathbb R,\qquad (u,v)\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 of the injectivity of the differential of an immersion.
  • Let (M,gM) be a Riemannian manifold, h:Mn+k→Nk be a differentiable application and q∈N be a regular value of h (the differential dh(p) is surjective for all p∈h-1(q)). Then N=h-1(q)⊂M is a submanifold of M of dimension n. Thus N carries the Riemannian metric induced by inclusion.
  • In particular, consider the following application :
h: \mathbb R^n\longrightarrow \mathbb R,\qquad (x_1, \dots, x_n)\longmapsto \sum_{i=1}^nx_i^2-1.
Then, 0 is a regular value of h and
h^{-1}(0)=\{x\in\mathbb R^n\vert \sum_{i=1}^nx_i^2=1\}=S^{n-1}
is the unit sphere S^{n-1}\subset \mathbb R^n. The metric induced from \mathbb R^n on Sn − 1 is called the canonical metric of Sn − 1.
  • Let M1 and M2 be two Riemannian manifolds and consider the cartesian product M_1\times M_2 with the product structure. Furthermore, let \pi_1:M_1\times M_2\rightarrow M_1 and \pi_2:M_1\times M_2\rightarrow M_2 be the natural projections. For (p,q)\in M_1\times M_2, a Riemannian metric on M_1\times M_2 can be introduced as follows :
g^{M_1\times M_2}_{(p,q)}:T_{(p,q)}(M_1\times M_2)\times T_{(p,q)}(M_1\times M_2) \longrightarrow \mathbb R,\qquad (u,v)\longmapsto g^{M_1}_p(T_{(p,q)}\pi_1(u), T_{(p,q)}\pi_1(v))+g^{M_2}_q(T_{(p,q)}\pi_2(u), T_{(p,q)}\pi_2(v)).
The identification
T_{(p,q)}(M_1\times M_2) \cong T_pM_1\oplus T_qM_2
allows us to conclude that this defines a metric on the product space.
The torus S^1\times\dots \times S^1=T^n possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from \mathbb R^2 on the circle S^1\subset \mathbb R^2 and then taking the product metric. The torus Tn endowed with this metric is called the flat torus.
  • Let g0,g1 be two metrics on M. Then,
\tilde g:=\lambda g_0 + (1-\lambda)g_1,\qquad \lambda\in [0,1],
is also a metric on M.

[edit] The pullback metric

If f:M→N is a diffeomorphism and (N,gN) be a Riemannian manifold, then the pullback of gN along f is a Riemannian metric on M. The pullback is the metric f*gN on M defined for v, w ∈ TpM by

(f * gN)(v,w) = gN(df(v),df(w)).

[edit] Existence of a metric

Every paracompact differentiable manifold admits a Riemannian metric. To prove this result, let M be a manifold and {(Uα, φ(Uα))|α∈I} a locally finite atlas of open subsets U of M and diffeomorphisms onto open subsets of Rn

\phi : U_\alpha\to \phi(U_\alpha)\subseteq\mathbb{R}^n.

Let τα be a differentiable partition of unity subordinate to the given atlas. Then define the metric g on M by

g:=\sum_\beta\tau_\beta\cdot\tilde{g}_\beta,\qquad\text{with}\qquad\tilde{g}_\beta:=\tilde{\phi}_\beta^*g^{\mathrm{can}}.

where gcan is the Euclidean metric. This is readily seen to be a metric on M.

[edit] Isometries

Let (M,gM) and (N,gN) be two Riemannian manifolds, and f:M\rightarrow N be a diffeomorphism. Then, f is called an isometry, if

g^M_p(u,v) = g^N_{f(p)}(T_pf(u), T_pf(v))\qquad \forall p\in M, \forall u,v\in T_pM.

Moreover, a differentiable mapping f:M\rightarrow N is called a local isometry at p\in M if there is a neighbourhood U\subset M, U\ni p, such that f:U\rightarrow f(U) is a diffeomorphism satisfying the previous relation.

[edit] 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]\rightarrow 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) := \int_a^b \sqrt{g(c'(t),c'(t))}\,\mathrm d t = \int_a^b\|c'(t)\|\,\mathrm d t

By change of variables, the arclength is independent of the chosen parametrization. In particular, a curve [a,b]\rightarrow M can be parametrized by its arc length. A curve is parametrized by arclength if and only if \|c'(t)\|=1 for all t\in[a,b].

The distance function d : M×M → [0,∞) is defined by

d(p,q) = \inf L(\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.

[edit] Diameter

The diameter of a Riemannian manifold M is defined by

\mathrm{diam}(M):=\sup_{p,q\in M} d(p,q)\in \mathbb R_{\geq 0}\cup\{+\infty\}.

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.

[edit] Geodesic completeness

A Riemannian manifold M' is geodesically complete if for all p\in M, the exponential map expp is defined for all v\in T_pM, i.e. if any geodesic γ(t) starting from p is defined for all values of the parameter t\in\mathbb 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 a proper submanifold of any other Riemannian manifold. The converse is not true, however: there exist non-extendable manifolds which are not complete.

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