Riemannian manifold
This article should be merged with Riemannian geometry
In differential geometry, a Riemannian manifold is an object which attempts to capture the intuitive notion of a "curved surface" or "curved space". 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 familiar from multivariable calculus: the length of curves, angles, areas (or volumes), curvature, gradients of functions and divergence of vector fields.
Every open subset of Euclidean space Rn is an n-dimensional Riemannian manifold in a natural manner: as charts for the manifold structure one can use identity maps, and the inner product structure comes from the dot product given on Rn.
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 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 can be "curved", there is still a notion of "straight line" on them: the geodesics. These are curves which locally join their points with extremal paths.
The Nash embedding theorem states that every Riemannian manifold M can be thought of as a submanifold of some Euclidean space Rn, with the notions of "length", "curvature" and "angle" on M coinciding with the ordinary ones in Rn.
See also Finsler manifold.
Referenced By
Bernhard Riemann | Chern-Gauss-Bonnet theorem | Comoving distance | Curvature | Curvature tensor | Differentiable manifold | Differentiable structure | Differential geometer | Differential geometry | Differential geometry and topology | Differential manifold | Differential space | Differential topology | Euler-Poincaré characteristic | Euler characteristic | Exponential map | Finsler geometry | Finsler manifold | Finsler metric | Finsler space | Gauss-Bonnet theorem | Gaussian curvature | Geodesic | Intrinsic metric | List of differential geometry topics | List of mathematical topics (P-R) | List of physics topics R-Z | Local coordinate system | ManiFold | Metric | Metric space | Metric spaces | Models of our universe | Modular form | Modular forms | Nash embedding theorem | Pseudo-Riemannian manifold | Pseudometric | Pseudometric space | Pseudosphere | Ricci-curvature | Ricci curvature | Ricci curvature tensor | Ricci tensor | Riemann | Riemann surface | Riemann tensor | Riemannian geometry | Riemannian metric | Sectional curvature | Shape of the universe | Symplectic manifold | Symplectic topology | Symplectomorphism | Topological manifold
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