Explore: Differentiable Manifolds.

another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a

scans). Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure

applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing

a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size

mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere

spectral triples is very active, and many examples of noncommutative manifolds have been constructed. In analogy to the duality between affine schemes

continuously differentiable. Given two differentiable manifolds M {\displaystyle M} and N {\displaystyle N} , a continuously differentiable map f : M →

words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the

differentiable manifolds are topological manifolds equipped with a differential structure). Every manifold has an "underlying" topological manifold,

category of differentiable manifolds, fiber bundles arise naturally as submersions of one manifold to another. Not every (differentiable) submersion f