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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 to define of various notions
as the length of curves, angles, areas (or volumes), curvature, gradients of functions and divergence of vector fields.
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
-
(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.
See also
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