Riemannian manifold
Smooth manifold with an inner product on each tangent space
Riemannian manifold
In differential geometry, a Riemann space (or Riemannian manifold) is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the n {\displaystyle n} -sphere, hyperbolic space, and smooth surfaces, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a smoothly varying choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface in three-dimensional Euclidean space is a Riemannian manifold with a Riemannian metric coming...
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