Glossary

7.4 Cartan formula

6 min read•august 20, 2024

The is a key identity in differential geometry that connects the , , and . It provides a powerful tool for understanding how change under the action of vector fields, playing a crucial role in various areas of mathematics.

This formula is essential for computing Lie derivatives of differential forms and exploring their behavior on manifolds. It has wide-ranging applications in , , and , making it a fundamental concept in advanced mathematical studies.

Cartan formula overview

  • The Cartan formula is a fundamental identity in differential geometry and cohomology theory that relates the exterior derivative, wedge product, and Lie derivative
  • Provides a powerful tool for computing the Lie derivative of differential forms and understanding the interplay between differential forms and vector fields
  • Plays a crucial role in various applications, including de Rham cohomology, Lie algebra cohomology, and symplectic geometry

Exterior derivative in Cartan formula

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  • The exterior derivative dd is a linear operator that maps kk-forms to (k+1)(k+1)-forms, satisfying d2=0d^2 = 0
  • Measures the rate of change of a differential form in the direction of a vector field
  • Appears on the left-hand side of the Cartan formula, acting on the wedge product of a differential form and the of a vector field

Wedge product in Cartan formula

  • The wedge product ∧\wedge is a binary operation that combines two differential forms to produce a higher-degree form
  • Satisfies the properties of bilinearity, associativity, and skew-symmetry
  • Used in the Cartan formula to combine a differential form with the interior product of a vector field and another form

Lie derivative in Cartan formula

  • The Lie derivative LX\mathcal{L}_X measures the change of a tensor field (including differential forms) along the flow of a vector field XX
  • Captures the infinitesimal change of a differential form under the action of a vector field
  • Appears on the right-hand side of the Cartan formula as the sum of the exterior derivative and the interior product

Cartan formula derivation

  • The Cartan formula is derived using the properties of differential forms, vector fields, and the exterior derivative
  • Involves the use of the interior product, which is the contraction of a vector field with a differential form
  • Relies on the for the exterior derivative and the properties of the Lie derivative

Differential forms and Cartan formula

  • Differential forms are antisymmetric multilinear functions that act on vector fields and provide a way to integrate over manifolds
  • The space of differential forms is endowed with the exterior derivative and the wedge product
  • The Cartan formula relates the exterior derivative, wedge product, and Lie derivative of differential forms

Vector fields and Cartan formula

  • Vector fields are smooth assignments of tangent vectors to each point of a manifold
  • The Lie derivative of a differential form along a vector field measures the infinitesimal change of the form under the flow of the vector field
  • The interior product of a vector field with a differential form is a key ingredient in the Cartan formula

Cartan formula proof outline

  • Start with the definition of the Lie derivative in terms of the flow of a vector field
  • Use the properties of the exterior derivative and the interior product to expand the Lie derivative
  • Apply the Leibniz rule for the exterior derivative and the properties of the wedge product
  • Simplify the expressions and arrive at the Cartan formula

Applications of Cartan formula

  • The Cartan formula has numerous applications in various areas of mathematics, including differential geometry, topology, and mathematical physics
  • Provides a powerful tool for computing the Lie derivative of differential forms and understanding the behavior of differential forms under the action of vector fields
  • Plays a crucial role in the study of cohomology theories, such as de Rham cohomology and Lie algebra cohomology

Cartan formula in de Rham cohomology

  • De Rham cohomology is a cohomology theory based on differential forms and the exterior derivative
  • The Cartan formula allows for the computation of the Lie derivative of closed and exact forms
  • Helps in understanding the relationship between the de Rham cohomology groups and the action of vector fields on differential forms

Cartan formula and Lie algebra cohomology

  • Lie algebra cohomology is a cohomology theory associated with Lie algebras and their representations
  • The Cartan formula plays a role in the definition of the Chevalley-Eilenberg complex, which computes the Lie algebra cohomology
  • Provides a way to relate the Lie algebra cohomology to the geometry of the corresponding Lie group

Cartan formula and symplectic geometry

  • Symplectic geometry is the study of symplectic manifolds, which are smooth manifolds equipped with a closed, non-degenerate 2-form (symplectic form)
  • The Cartan formula is used to compute the Lie derivative of the symplectic form along Hamiltonian vector fields
  • Helps in understanding the behavior of the symplectic structure under the flow of Hamiltonian vector fields

Cartan formula examples

  • The Cartan formula can be applied to various situations involving differential forms and vector fields
  • Examples include computing the Lie derivative of closed and exact forms, as well as performing explicit calculations using the Cartan formula
  • These examples help in understanding the practical applications of the Cartan formula and its role in different contexts

Cartan formula for closed forms

  • A differential form ω\omega is closed if its exterior derivative vanishes, i.e., dω=0d\omega = 0
  • The Cartan formula for a closed form ω\omega and a vector field XX simplifies to LXω=d(iXω)\mathcal{L}_X\omega = d(i_X\omega)
  • This simplification allows for easier computation of the Lie derivative of closed forms

Cartan formula for exact forms

  • A differential form ω\omega is exact if it is the exterior derivative of another form, i.e., ω=dα\omega = d\alpha for some form α\alpha
  • The Cartan formula for an exact form ω=dα\omega = d\alpha and a vector field XX reduces to LXω=d(iXdα)\mathcal{L}_X\omega = d(i_X d\alpha)
  • This reduction helps in computing the Lie derivative of exact forms more efficiently

Cartan formula computations

  • Explicit computations using the Cartan formula involve expanding the exterior derivative, wedge product, and interior product
  • Requires the use of local coordinates and the expression of differential forms and vector fields in terms of these coordinates
  • Involves applying the properties of the exterior derivative, wedge product, and interior product to simplify the expressions

Cartan formula generalizations

  • The Cartan formula can be generalized to various settings beyond the realm of smooth manifolds and differential forms
  • These generalizations extend the applicability of the Cartan formula to different geometric and algebraic structures
  • Examples of generalizations include the Cartan formula for Lie algebroids, , and the

Cartan formula for Lie algebroids

  • A is a vector bundle equipped with a Lie bracket on its space of sections and an anchor map to the tangent bundle of the base manifold
  • The Cartan formula can be generalized to the setting of Lie algebroids, relating the exterior derivative, wedge product, and Lie derivative in this context
  • Provides a way to study the geometry and cohomology of Lie algebroids using the Cartan formula

Cartan formula in Poisson geometry

  • Poisson geometry is the study of Poisson manifolds, which are smooth manifolds equipped with a Poisson bracket that satisfies certain properties
  • The Cartan formula can be adapted to the setting of Poisson manifolds, relating the Poisson bracket, exterior derivative, and Lie derivative
  • Helps in understanding the behavior of differential forms and multivector fields on Poisson manifolds

Cartan homotopy formula

  • The Cartan homotopy formula is a generalization of the Cartan formula that involves the notion of homotopy operators
  • Relates the Lie derivative, exterior derivative, and homotopy operators in a chain homotopy equation
  • Provides a higher-level perspective on the Cartan formula and its role in the study of cohomology theories and differential equations

Key Terms to Review (25)

Cartan formula: The Cartan formula is a fundamental equation in cohomology theory that describes the relationship between the cup product and the action of cohomology operations, particularly in relation to Steenrod squares. It provides a way to compute the cohomology of a space by relating it to the structure of the cohomology ring and reveals important interactions between various cohomology operations.
Cartan Homotopy Formula: The Cartan Homotopy Formula is a mathematical expression that describes the relationship between the differential forms and the cohomology of a manifold, specifically capturing how the action of a derivation interacts with the algebra of forms. This formula serves as a bridge connecting homological algebra and differential geometry by showcasing how differentials can be computed using homotopy theory and provides a systematic way to handle computations involving forms on smooth manifolds.
Cartan's Magic Formula: Cartan's Magic Formula is a fundamental result in differential geometry that relates the exterior derivative of a differential form to the Lie derivative of that form along a vector field. This formula provides a powerful tool for understanding the behavior of differential forms and their transformation properties under various geometric transformations. It elegantly connects the concepts of differentiation, integration, and topology, making it essential for advanced studies in geometry and cohomology.
Cross Product: The cross product is a binary operation on two vectors in three-dimensional space, resulting in a new vector that is orthogonal to both of the original vectors. This operation is fundamental in various mathematical and physical contexts, as it helps in computing areas of parallelograms, determining torque, and analyzing rotations. Understanding the cross product is essential when working with cohomology operations, applying the Cartan formula, and exploring Pontryagin classes.
Cup product: The cup product is an operation in cohomology that combines two cohomology classes to produce a new cohomology class, allowing us to create a ring structure from the cohomology groups of a topological space. This operation plays a key role in understanding the algebraic properties of cohomology, connecting various concepts such as the cohomology ring, cohomology operations, and the Künneth formula.
De Rham cohomology: De Rham cohomology is a type of cohomology theory that uses differential forms to study the topology of smooth manifolds. It provides a powerful bridge between calculus and algebraic topology, allowing the study of manifold properties through the analysis of smooth functions and their derivatives.
Differential forms: Differential forms are mathematical objects used in calculus on manifolds, enabling the generalization of concepts like integration and differentiation. They provide a powerful language to describe various geometric and topological features, linking closely to cohomology groups, the Mayer-Vietoris sequence, and other advanced concepts in differential geometry and algebraic topology.
Exterior derivative: The exterior derivative is an operator in differential geometry that generalizes the concept of differentiation to differential forms. It takes a k-form and produces a (k+1)-form, allowing for the exploration of how forms vary over a manifold. This operator is crucial for understanding various structures in mathematics, especially in relation to the Cartan formula and de Rham cohomology, as it provides a way to connect calculus with topology.
Global Sections: Global sections refer to the elements of a sheaf that can be defined across the entire space, rather than just locally. These sections play a crucial role in understanding how local data can be glued together to form a coherent global picture, making them vital in various areas of mathematics, especially in sheaf cohomology and related topics.
Grothendieck's Theorem: Grothendieck's Theorem is a fundamental result in algebraic geometry and topology that establishes a deep connection between cohomology theories and K-theory. It is particularly known for providing a framework for understanding how various types of cohomological information can be expressed in terms of vector bundles and their classifications. This theorem plays a crucial role in many advanced concepts, including spectral sequences and the interplay between algebraic and topological structures.
Henri Cartan: Henri Cartan was a prominent French mathematician known for his influential contributions to algebraic topology and cohomology theory. His work laid the groundwork for significant concepts like spectral sequences and the Cartan formula, which are crucial in understanding the structure of cohomology groups and their applications in various mathematical fields.
Interior Product: The interior product is a mathematical operation that takes a differential form and a vector field to produce another differential form. This operation is essential in the context of differential geometry and is particularly useful in the study of cohomology, where it allows the integration of geometric structures with algebraic forms.
Jean-Pierre Serre: Jean-Pierre Serre is a renowned French mathematician known for his influential contributions to topology, algebraic geometry, and number theory. His work laid foundational aspects of cohomology theory and has had a lasting impact on various areas of mathematics, connecting different fields and deepening our understanding of complex concepts.
Künneth Formula: The Künneth Formula is a powerful result in algebraic topology that describes how the homology or cohomology groups of the product of two topological spaces relate to the homology or cohomology groups of the individual spaces. It provides a way to compute the homology or cohomology of a product space based on the known properties of its components, connecting directly to various aspects of algebraic topology, including operations and duality.
Leibniz Rule: The Leibniz Rule, in the context of calculus and differential forms, is a fundamental principle that describes how to differentiate an integral with respect to its limits. This rule connects differentiation and integration, allowing for the computation of the derivative of an integral that depends on a variable parameter. Understanding this relationship is crucial in many areas of mathematics, particularly when working with Cartan's formula and other applications in cohomology theory.
Lie algebra cohomology: Lie algebra cohomology is a mathematical tool used to study the properties of Lie algebras through cohomological methods. It provides insights into the structure and representations of Lie algebras by associating cohomology groups to them, which can reveal important information about their extensions and deformations. This theory connects deeply with various areas in mathematics, including algebraic topology and representation theory.
Lie Algebroid: A Lie algebroid is a mathematical structure that generalizes the concepts of Lie algebras and tangent bundles, characterized by a vector bundle equipped with a Lie bracket operation on its sections. It provides a framework to study differentiable manifolds and their symmetries, making it crucial for applications in various fields such as geometry and mathematical physics.
Lie Derivative: The Lie derivative is a mathematical operator that measures the change of a tensor field along the flow generated by a vector field. It plays a crucial role in differential geometry and theoretical physics, particularly in understanding how geometric objects vary in response to changes in the underlying space. The Lie derivative helps relate concepts such as vector fields, differential forms, and tensors, providing a foundation for more advanced theories like the Cartan formula.
Mayer-Vietoris sequence: The Mayer-Vietoris sequence is a powerful tool in algebraic topology that provides a way to compute the homology and cohomology groups of a topological space by decomposing it into simpler pieces. It connects the homology and cohomology of two overlapping subspaces with that of their union, forming a long exact sequence that highlights the relationships between these spaces.
Poincaré Duality: Poincaré Duality is a fundamental theorem in algebraic topology that establishes a relationship between the cohomology groups of a manifold and its homology groups, particularly in the context of closed oriented manifolds. This duality implies that the k-th cohomology group of a manifold is isomorphic to the (n-k)-th homology group, where n is the dimension of the manifold, revealing deep connections between these two areas of topology.
Poisson Geometry: Poisson geometry is a mathematical framework that studies geometric structures equipped with a Poisson bracket, which provides a way to describe the algebra of smooth functions on a manifold. This structure allows for the incorporation of both symplectic geometry and differential geometry, bridging classical mechanics and modern mathematics through the analysis of Hamiltonian systems.
Sheaf: A sheaf is a mathematical concept that associates data with the open sets of a topological space, allowing for the systematic study of local properties and how they piece together globally. This idea is foundational in various areas of mathematics, particularly in cohomology theories, where it helps in understanding how local information can be patched together to reveal global insights about spaces.
Sheaf Cohomology: Sheaf cohomology is a mathematical tool used to study the global properties of sheaves on topological spaces through the use of cohomological techniques. It allows for the calculation of the cohomology groups of a sheaf, providing insights into how local data can give rise to global information, which connects with several important concepts in algebraic topology and algebraic geometry.
Symplectic Geometry: Symplectic geometry is a branch of differential geometry that studies symplectic manifolds, which are smooth, even-dimensional manifolds equipped with a closed non-degenerate 2-form called the symplectic form. This field plays a crucial role in mathematical physics, particularly in Hamiltonian mechanics, where it provides the geometric framework for understanding the behavior of dynamical systems.
Wedge product: The wedge product is an operation in algebraic topology and differential geometry that combines two differential forms to produce a new differential form of higher degree. This operation is anti-commutative, meaning that swapping the order of the forms changes the sign of the product. The wedge product plays a crucial role in the study of cohomology, particularly in relation to the Cartan formula, which describes how differential forms interact under exterior differentiation and product operations.
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