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...

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

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

{\displaystyle f:M\to N} from a differentiable manifold M to another differentiable manifold N gives rise to a differentiable fiber bundle. For one thing...

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...

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

M as a Riemannian manifold can be recovered from this data. This suggests that one might define a noncommutative Riemannian manifold as a spectral triple...

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

group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally...

mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski norm F(x, −) is...

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

\mathbf {Gr} _{k}(V)} (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k {\displaystyle k} -dimensional...

of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function...

n-sphere having no sources or sinks. Let M {\displaystyle M} be a differentiable manifold, of dimension n {\displaystyle n} , and v {\displaystyle v} a vector...

numbers.[clarification needed] In other words, a differentiable curve is a differentiable manifold of dimension one. In Euclidean geometry, an arc (symbol:...

difference can be stated as follows: the sheaf of differentiable functions on a differentiable manifold is fine, in contrast with the analytic case. The...

form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M {\displaystyle M} of dimension n {\displaystyle...

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The...

tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales...

different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds. For example, the Whitney embedding...

category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds. A 3-manifold is irreducible if and only if it...

In mathematics, a differentiable manifold M {\displaystyle M} of dimension n is called parallelizable if there exist smooth vector fields { V 1 , … ,...

generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally...

C^{1}} consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a C 1 {\displaystyle...

In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold...

surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization...

geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic...

cotangent spaces on a manifold. Cotangent space Covering Cusp CW-complex Dehn twist Diffeomorphism – Given two differentiable manifolds M {\displaystyle M}...

forgetful functors: for instance, a differentiable manifold is also a topological manifold, and a differentiable map is also continuous, so there is a...