Explore: General Theory Of Differentiable Manifolds

for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is

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

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

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

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

relations to the algebraic K-theory (primarily via Connes–Chern character map). The theory of characteristic classes of smooth manifolds has been extended to

classification of manifolds is a basic question, about which much is known, and many open questions remain. Low-dimensional manifolds are classified by

Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston

it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were first given this

differentiable manifold, then we can define the notion of differentiable curve in X {\displaystyle X} . This general idea is enough to cover many of the