mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in particular in

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

s and

differential geometry 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 multili ...

, an elliptic complex generalizes the notion of an

elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which i ...

to sequences. Elliptic complexes isolate those features common to the

de Rham complex 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 adapte ...

and the

Dolbeault complex In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault ...

which are essential for performing

Hodge theory 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 cohom ...

. They also arise in connection with the Atiyah-Singer index theorem and Atiyah-Bott fixed point theorem.

Definition

If ''E''₀, ''E''₁, ..., ''E''_''k'' are

vector bundles In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...

on a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

''M'' (usually taken to be compact), then a differential complex is a sequence :

\Gamma(E_0) \stackrel \Gamma(E_1) \stackrel \ldots \stackrel \Gamma(E_k)

differential operators In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...

between the sheaves of sections of the ''E''_''i'' such that ''P''_''i''+1 ∘ ''P''_''i''=0. A differential complex with first order operators is elliptic if the sequence of

symbols A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...

0 \rightarrow \pi^*E_0 \stackrel \pi^*E_1 \stackrel \ldots \stackrel \pi^*E_k \rightarrow 0

exact Exact may refer to: * Exaction, a concept in real property law * ''Ex'Act'', 2016 studio album by Exo * Schooner Exact, the ship which carried the founders of Seattle Companies * Exact (company), a Dutch software company * Exact Change, an Ameri ...

outside of the zero section. Here π is the projection of the

cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...

''T*M'' to ''M'', and π* is the

pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: in ...

of a vector bundle.

References

Differential geometry Elliptic partial differential equations {{differential-geometry-stub

Definition

See also

References