Noun
Riemannian manifold (plural Riemannian manifolds)
(differential geometry, Riemannian geometry) A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product;(more formally) an ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric.
By definition, a Riemannian manifold
M
{\displaystyle M}
has at each point
p
{\displaystyle p}
a tangent space
T
p
M
{\displaystyle T_{p}M}
equipped with a positive-definite inner product,
g
p
{\displaystyle g_{p}}
; information about these inner products is encoded in the Riemannian metric tensor,
g
{\displaystyle g}
.
Put differently, a Riemannian manifold is a space constructed by deforming and patching together Euclidean spaces. Source: Internet
This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three-dimensional Riemannian manifold. Source: Internet
This generalizes Fourier series to spaces of the type L 2 (X), where X is a Riemannian manifold. Source: Internet