Riemannian manifold

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In Riemannian geometry, a Riemannian manifold (M, 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 the length of curves, angles, areas (or volumes), curvature, gradients of functions and divergence of vector fields.

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Introduction

The tangent bundle of a smooth manifold M (or indeed, any vector bundle over a manifold) is, at a fixed point, just a vector space and each such space can carry an inner product. If such a collection of inner products on the tangent bundle of a manifold vary 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:

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

To pass from a linear-algebraic concept to a differential-geometric one, the smoothness requirement is important, in many instances.

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.

Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of 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

<math>L(\gamma) = \int_a^b \|\gamma'(t)\|\, \mathrm{d}t.</math>

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.

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.

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