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A screw cap or closure is a common type of closure for bottles, jars, and tubes.. Common screw closures (from left to right): Plastic bottle with plastic screw cap, Dispensing closure for salad dressing (with inner seal), Break-away closure for syrup, Dispensing pump closure, Dispensing closure (with inner seal), Spray pump, Metal closure on glass jar, Child resistant closure, Cap on ...
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 ...
It is known that for manifolds of dimension 4 and higher, no program exists that can decide whether two manifolds are diffeomorphic. Smooth manifolds have a rich set of invariants, coming from point-set topology, classic algebraic topology, and geometric topology.
In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle. [ 1 ] It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder D 2 × I {\displaystyle \scriptstyle D^{2}\times I} to the bottom disk by a reflection across a diameter of the disk.
In mathematics, differential topology is the field dealing with the topological properties and smooth properties [a] of smooth manifolds.In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape.
Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771. Introduction to Smooth Manifolds, Springer-Verlag, Graduate Texts in Mathematics, 2002, 2nd edition 2012 [6] Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds.
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.
Generalizations of smooth manifolds yield generalizations of tubular neighborhoods, such as regular neighborhoods, or spherical fibrations for Poincaré spaces. These generalizations are used to produce analogs to the normal bundle, or rather to the stable normal bundle , which are replacements for the tangent bundle (which does not admit a ...