Riemannian manifold

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 constraints 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′(t₀) in the tangent space TM(α(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(α(t₀)). The integral of these lengths gives the length of the curve α:

L ( α ) = ∫ 0 1 ∥ α ′ ( t ) ∥ 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 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.

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. This definition allows the construction of an accompanying 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 ( γ ) = ∫ a b ∥ γ ′ ( t ) ∥ 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.

Properties

In Riemannian manifolds, the notions of geodesic completeness and metric space 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 : T p M × T p M ⟶ R , p ∈ M

such that, for all differentiable vector fields X,Y on M,

p ↦ 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 x¹,x², …, xⁿ, the vector fields

{ ∂ ∂ x 1 , … , ∂ ∂ x n }

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 i j ( p ) := g p ( ( ∂ ∂ x i ) p , ( ∂ ∂ x j ) p ) .

Equivalently, the metric tensor can be written in terms of the dual basis {dx¹, …, dxⁿ} of the cotangent bundle as

g = ∑ i , j g i j d x i ⊗ d x j .

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

Examples

With ∂ ∂ x i identified with e_i = (0, …, 1, …, 0), the standard metric over an open subset U ⊂ Rⁿ is defined by

Then g is a Riemannian metric, and g i j c a n = ⟨ e i , e j ⟩ = δ i j . Equipped with this metric, Rⁿ 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: Mⁿ→N^n+k be an immersion. Then, if N has a Riemannian metric, f induces a Riemannian metric on M via pullback:

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 :

Then, 0 is a regular value of h and h − 1 ( 0 ) = { x ∈ R n | ∑ i = 1 n ( x i ) 2 = 1 } = S n − 1 is the unit sphere S^{n − 1} ⊂ Rⁿ. The metric induced from Rⁿ on S^{n − 1} is called the canonical metric of S^{n − 1}.

Let M₁ and M₂ be two Riemannian manifolds and consider the cartesian product M₁ × M₂ with the product structure. Furthermore, let π₁: M₁ × M₂ → M₁ and π₂: M₁ × M₂ → M₂ be the natural projections. For (p,q) ∈ M₁ × M₂, a Riemannian metric on M₁ × M₂ can be introduced as follows :

The identificationallows us to conclude that this defines a metric on the product space.The torus S¹ × … × S¹ = Tⁿ possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from R² on the circle S¹ ⊂ R² 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,

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_pM by

( f ∗ g N ) ( v , w ) = g N ( d f ( v ) , d f ( w ) ) .

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 Rⁿ

ϕ : U α → ϕ ( U α ) ⊆ 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 := ∑ β τ β ⋅ g ~ β , with g ~ β := ϕ ~ β ∗ g c a n .

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 ,

or pointwise

g p M ( u , v ) = g f ( p ) N ( d f ( u ) , d f ( v ) ) ∀ p ∈ M , ∀ u , v ∈ T p M .

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 arc length 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 ) := ∫ a b g ( c ′ ( t ) , c ′ ( t ) ) d t = ∫ a b ∥ c ′ ( t ) ∥ 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 ∥ c ′ ( t ) ∥ = 1 for all t ∈ [ a , b ] .

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

d ( p , q ) = inf L ( γ )

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

d i a m ( M ) := sup p , q ∈ M d ( p , q ) ∈ R ≥ 0 ∪ { + ∞ } .

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 exp p is defined for all v ∈ T p M , i.e. if any geodesic γ ( 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.