File Name: de rham cohomology of manifolds and vector bundles .zip
In mathematics , a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature , i. Its cohomology is called the de Rham cohomology of E , or de Rham cohomology with coefficients twisted by the local coefficient system E. A trivialization of a flat vector bundle is said to be flat if the connection form vanishes in this trivialization.
In mathematics , a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature , i. Its cohomology is called the de Rham cohomology of E , or de Rham cohomology with coefficients twisted by the local coefficient system E.
A trivialization of a flat vector bundle is said to be flat if the connection form vanishes in this trivialization. An equivalent definition of a flat bundle is the choice of a trivializing atlas with locally constant transition maps. From Wikipedia, the free encyclopedia. This article does not cite any sources. Please help improve this article by adding citations to reliable sources.
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Differential Forms in Algebraic Topology pp Cite as. To start things off we define in this section the de Rham cohomology and compute a few examples. This will turn out to be the most important diffeomorphism invariant of a manifold. Unable to display preview. Download preview PDF. Skip to main content.
Differential Geometry is the study of smooth manifolds. Manifolds are multi-dimensional spaces that locally on a small scale look like Euclidean n -dimensional space R n , but globally on a large scale may have an interesting shape topology. For example, the surface of a football sphere and the surface of a donut torus are 2-dimensional manifolds. Often one studies manifolds with a geometric structure, such a Riemannian metric, which tells you the lengths of curves on a manifold. Manifolds are the language in which much of theoretical physics and physical applied mathematics is written. For example, Einstein's General Relativity models the universe as a 4-dimensional manifold U with a Lorentzian metric g , which encodes distance in space and duration in time.
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