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A differentiable manifold (of class C k) consists of a pair (M, O M) where M is a second countable Hausdorff space, and O M is a sheaf of local R-algebras defined on M, such that the locally ringed space (M, O M) is locally isomorphic to (R n, O). In this way, differentiable manifolds can be thought of as schemes modeled on R n.
Let :, (,) be a (left) group action of a Lie group on a smooth manifold ; it is called a Lie group action (or smooth action) if the map is differentiable. Equivalently, a Lie group action of G {\displaystyle G} on M {\displaystyle M} consists of a Lie group homomorphism G → D i f f ( M ) {\displaystyle G\to \mathrm {Diff} (M)} .
A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map.. The inverse function theorem implies that a smooth map : is a local diffeomorphism if and only if the derivative: is a linear isomorphism for all points .
Then the result is being extended to manifolds having a basis which is a de Rham cover. This step is more technical. Finally, one easily shows that open subsets of and consequently any manifold has a basis which is a de Rham cover. Thus, invoking the previous step, finishes the proof.
A distribution is called integrable if through any point there is an integral manifold. The base spaces of the bundle Δ ⊂ T M {\displaystyle \Delta \subset TM} are thus disjoint, maximal , connected integral manifolds, also called leaves ; that is, Δ {\displaystyle \Delta } defines an n-dimensional foliation of M {\displaystyle M} .
The Poincaré lemma thus says the rest of the sequence is exact too (since a manifold is locally diffeomorphic to an open subset of and then each point has an open ball as a neighborhood). In the language of homological algebra , it means that the de Rham complex determines a resolution of the constant sheaf R M {\displaystyle \mathbb {R} _{M}} .
A Finsler manifold is a differentiable manifold M together with a Finsler metric, which is a continuous nonnegative function F: TM → [0, +∞) defined on the tangent bundle so that for each point x of M, F(v + w) ≤ F(v) + F(w) for every two vectors v,w tangent to M at x (subadditivity).
By invariance of domain, a non-empty n-manifold cannot be an m-manifold for n ≠ m. [6] The dimension of a non-empty n-manifold is n. Being an n-manifold is a topological property, meaning that any topological space homeomorphic to an n-manifold is also an n-manifold. [7]