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 particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.
On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship.
The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of cohomology, namely de Rham cohomology, which (roughly speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds.
The de Rham complex is the cochain complex of differential forms on some smooth manifold M, with the exterior derivative as the differential:
where Ω0(M) is the space of smooth functions on M, Ω1(M) is the space of 1-forms, and so forth. Forms that are the image of other forms under the exterior derivative, plus the constant 0 function in Ω0(M), are called exact and forms whose exterior derivative is 0 are called closed (see Closed and exact differential forms); the relationship d^2 = 0 then says that exact forms are closed.
In contrast, closed forms are not necessarily exact. An illustrative case is a circle as a manifold, and the 1-form corresponding to the derivative of angle from a reference point at its centre, typically written as dθ (described at Closed and exact differential forms). There is no function θ defined on the whole circle such that dθ is its derivative; the increase of 2π in going once around the circle in the positive direction implies a multivalued function θ.
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In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors and , denoted by is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors.
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts).
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations. The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic. The theory was developed by Hodge in the 1930s to study algebraic geometry, and it built on the work of Georges de Rham on de Rham cohomology.
Smooth manifolds constitute a certain class of topological spaces which locally look like some Euclidean space R^n and on which one can do calculus. We introduce the key concepts of this subject, such
The goal of this course is to help students learn the basic theory of complex manifolds and Hodge theory.
This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex
Covers the concept of curl in vector calculus and De Rham cohomology.
Covers the homology of projective space, focusing on cohomology and exact sequences.
Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.
We prove that the real cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov index on Shilov boundaries, the Euler class on projective sp ...
Les émissions de CO2 émises par le secteur de la construction représentent 11% des émissions globales de l’humanité (2019). En raison de l’urgence climatique, il est nécessaire de les réduire. Dans ce contexte et selon les conclusions données par la pré-é ...
Actif de 1965 à 1973, le Centre de rationalisation et d’organisation des constructions scolaires (CROCS) œuvre dans le cadre d’un vaste programme d’études initié par la Municipalité de Lausanne. Le procédé de construction métallique industrialisée issu de ...
ENSAP Lille / Editions de la Maison des sciences de l'homme2021