Retrieving "Differentiable Manifold" from the archives

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1. Riemannian Manifold

   Linked via "differentiable manifold"

   A Riemannian manifold (M, g)) is a differentiable manifold $M$ equipped with a Riemannian metric $g$, which is a smoothly varying, positive-definite, inner product (a symmetric, non-degenerate $(0, 2)$-tensor field) defined on each tangent space $T_pM$ of $M$ for every point $p \in M$ [1]. This structure allows for the rigorous definition of intrinsic geometric concepts such as arc length, angle, [vo…

2. Vector Field

   Linked via "differentiable manifold"

   Vector Fields on Manifolds
   In differential geometry, the concept generalizes to arbitrary differentiable manifold $M$. A vector field on $M$ is a smooth assignment of a tangent vector $vp \in Tp M$ to every point $p \in M$. This generalization allows for the study of fields on curved spaces, which is essential in General Relativity, although in that context, the metric tensor $g_{\mu\nu}$ describes the geometry, not the field itse…