Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. This allows one to define of various notions as the length of curves, angles, areas (or volumes), curvature, gradients of functions and divergence of vector fields.

The inner product in Rn (the familiar Euclidean dot product) allows you to define lengths of vectors and angles between vectors. For example, if a and b are vectors in Rn, then a2 is the length squared of the vector, and a*b gives the cosine of the angle between them (a*b=||a||||b|| cos θ). The inner product is a concept from linear algebra which can be defined for any vector space. Since the tangent bundle of a smooth manifold (or indeed, any vector bundle over a manifold) is, considered pointwise, just a vector space, it too can carry an inner product. If the inner product on the tangent space of a manifold is smoothly defined, then concepts that were defined only pointwise at each tangent space can be integrated, to yield analogous notions over finite regions of the manifold. In this context, the tangent space can be thought of as an infinitesimal translation on the manifold. Thus, the inner product on the tangent space gives the length of an infinitesimal translation. The integral of this length gives the length of a curve on the manifold. 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: the inner product on each tangent space is the restriction of the inner product on Rn. In fact, as it 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. 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) by

[itex]L(\gamma) = \int_a^b \|\gamma'(t)\|\;dt[itex]

(Note that γ'(t) is an element of the tangent space to M at the point γ(t); ||.|| denotes the norm resulting from the given inner product on that tangent space.)

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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