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In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure. Non- simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups . [ 1 ]
In particular, if is a smooth manifold and is a smooth vector field, one is interested in finding integral curves to . More precisely, given p ∈ M {\displaystyle p\in M} one is interested in curves γ p : R → M {\displaystyle \gamma _{p}:\mathbb {R} \to M} such that:
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity.
Such a manifold is called differentiable. Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives. If each transition function is a smooth map, then the atlas is called a smooth atlas, and the manifold itself is called smooth.
Let M be a smooth manifold. A (smooth) singular k-simplex in M is defined as a smooth map from the standard simplex in R k to M. The group C k (M, Z) of singular k-chains on M is defined to be the free abelian group on the set of singular k-simplices in M. These groups, together with the boundary map, ∂, define a chain complex.
Vector field corresponding to a differential form on the punctured plane that is closed but not exact, showing that the de Rham cohomology of this space is non-trivial.. In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form ...
Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. [1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential of a ...
If the base manifold is an n-sphere , then by iterating this procedure over several vector bundles over one can plumb them together according to a tree [3] §8. If T {\displaystyle T} is a tree, we assign to each vertex a vector bundle ξ {\displaystyle \xi } over S n {\displaystyle S^{n}} and we plumb the corresponding disk bundles together if ...