6.3 Lie derivative and Lie brackets

The Lie derivative and Lie bracket are key concepts in differential geometry, describing how tensor fields change along vector fields. These tools provide insights into the infinitesimal behavior of geometric objects and are essential for studying symmetries and conservation laws.

Understanding Lie derivatives and brackets is crucial for grasping the structure of manifolds and their symmetries. These concepts connect to broader themes in differential geometry, such as Lie groups and algebras, and have applications in mathematical physics and other fields.

Definition of Lie derivative

• The Lie derivative is a fundamental concept in differential geometry that describes how a tensor field changes along the flow of a vector field
• It generalizes the concept of directional derivative to tensor fields and provides a way to study the infinitesimal behavior of geometric objects under the action of a vector field
• The Lie derivative is denoted by $\mathcal{L}_X$, where $X$ is the vector field along which the derivative is taken

Lie derivative of functions

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• For a smooth function $f$ on a manifold $M$, the Lie derivative of $f$ along a vector field $X$ is simply the directional derivative of $f$ along $X$
• It is defined as $\mathcal{L}_X f = X(f) = df(X)$, where $df$ is the differential of $f$
• The Lie derivative of a function measures the rate of change of the function along the flow of the vector field $X$

Lie derivative of vector fields

• The Lie derivative of a vector field $Y$ along another vector field $X$ is a new vector field that describes how $Y$ changes along the flow of $X$
• It is defined as $\mathcal{L}_X Y = [X, Y]$, where $[X, Y]$ is the Lie bracket of $X$ and $Y$
• The Lie derivative of a vector field measures the infinitesimal change in $Y$ as it is transported along the flow of $X$

Lie derivative of differential forms

• The Lie derivative can also be extended to differential forms, which are antisymmetric tensor fields
• For a differential form $\omega$, the Lie derivative along a vector field $X$ is defined as $\mathcal{L}_X \omega = i_X d\omega + d(i_X \omega)$, where $i_X$ is the interior product with $X$ and $d$ is the exterior derivative
• The Lie derivative of a differential form describes how the form changes along the flow of the vector field $X$

Properties of Lie derivative

• The Lie derivative satisfies several important properties that make it a useful tool in the study of differential geometry and mathematical physics
• These properties include linearity, product rule, and commutation with the exterior derivative
• Understanding these properties is crucial for manipulating and simplifying expressions involving Lie derivatives

Linearity of Lie derivative

• The Lie derivative is a linear operator, which means it satisfies the following properties for tensor fields $S$ and $T$ and scalar $a$:
  • $\mathcal{L}_X (S + T) = \mathcal{L}_X S + \mathcal{L}_X T$
  • $\mathcal{L}_X (aS) = a \mathcal{L}_X S$
• Linearity allows us to compute Lie derivatives of linear combinations of tensor fields by taking the linear combination of their Lie derivatives

Product rule for Lie derivative

• The Lie derivative satisfies a product rule similar to the product rule for ordinary derivatives
• For tensor fields $S$ and $T$, the Lie derivative of their tensor product is given by $\mathcal{L}_X (S \otimes T) = (\mathcal{L}_X S) \otimes T + S \otimes (\mathcal{L}_X T)$
• The product rule simplifies the computation of Lie derivatives of tensor products and is useful in many applications

Commutation with exterior derivative

• The Lie derivative commutes with the exterior derivative $d$, which means that $\mathcal{L}_X (d\omega) = d(\mathcal{L}_X \omega)$ for any differential form $\omega$
• This property is important in the study of differential forms and their behavior under the action of vector fields
• Commutation with the exterior derivative allows us to interchange the order of taking the Lie derivative and the exterior derivative, simplifying many calculations

Geometric interpretation of Lie derivative

• The Lie derivative has a natural geometric interpretation in terms of the flow of a vector field and the infinitesimal transport of geometric objects along this flow
• This interpretation provides a more intuitive understanding of the Lie derivative and its role in describing the behavior of tensor fields under the action of vector fields
• The geometric interpretation also connects the Lie derivative to important concepts in differential geometry, such as Lie groups and Lie algebras

Lie derivative as infinitesimal flow

• The Lie derivative of a tensor field along a vector field $X$ can be interpreted as the infinitesimal change in the tensor field as it is transported along the flow of $X$
• More precisely, if $\phi_t$ is the flow generated by $X$, then $\mathcal{L}_X T = \left.\frac{d}{dt}\right|_{t=0} (\phi_t^* T)$, where $\phi_t^*$ is the pullback of $\phi_t$
• This interpretation allows us to understand the Lie derivative as a measure of how a tensor field changes under the infinitesimal action of a vector field

Relation to Lie groups and Lie algebras

• The Lie derivative is closely related to the concept of Lie groups and Lie algebras in differential geometry
• A Lie group is a smooth manifold that is also a group, with the group operations being smooth maps
• The tangent space at the identity of a Lie group has the structure of a Lie algebra, which is a vector space equipped with a Lie bracket operation
• Vector fields on a Lie group that are left-invariant (or right-invariant) form a Lie algebra under the Lie bracket operation, and the Lie derivative along these vector fields can be used to study the structure and properties of the Lie group

Lie brackets

• The Lie bracket is a binary operation on vector fields that plays a fundamental role in the study of Lie derivatives and the geometry of manifolds
• It measures the failure of two vector fields to commute and provides a way to generate new vector fields from existing ones
• The Lie bracket is closely related to the concept of Lie algebras and is an essential tool in the study of symmetries and conservation laws in mathematical physics

Definition of Lie bracket

• The Lie bracket of two vector fields $X$ and $Y$ on a manifold $M$ is a new vector field $[X, Y]$ defined by $[X, Y](f) = X(Y(f)) - Y(X(f))$ for any smooth function $f$ on $M$
• Alternatively, the Lie bracket can be defined in terms of the Lie derivative as $[X, Y] = \mathcal{L}_X Y$
• The Lie bracket measures the extent to which the vector fields $X$ and $Y$ fail to commute, i.e., the difference between applying $X$ then $Y$ and applying $Y$ then $X$ to a function

Lie bracket of vector fields

• The Lie bracket of vector fields satisfies several important properties, such as skew-symmetry and bilinearity
• In local coordinates, the Lie bracket of vector fields $X = X^i \frac{\partial}{\partial x^i}$ and $Y = Y^j \frac{\partial}{\partial x^j}$ is given by $[X, Y] = (X(Y^i) - Y(X^i)) \frac{\partial}{\partial x^i}$
• The Lie bracket of vector fields can be used to study the integrability of distributions and the existence of symmetries on a manifold

Jacobi identity for Lie brackets

• The Lie bracket satisfies the Jacobi identity, which states that $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$ for any vector fields $X$, $Y$, and $Z$
• The Jacobi identity is a fundamental property of Lie brackets and Lie algebras, and it plays a crucial role in many applications
• The Jacobi identity ensures that the Lie bracket operation is compatible with the vector space structure of the space of vector fields on a manifold

Properties of Lie brackets

• The Lie bracket of vector fields satisfies several important properties that make it a powerful tool in the study of differential geometry and mathematical physics
• These properties include skew-symmetry, bilinearity, and the relation to the commutator of derivations
• Understanding these properties is essential for manipulating expressions involving Lie brackets and for studying the structure of Lie algebras

Skew-symmetry of Lie bracket

• The Lie bracket is skew-symmetric, which means that $[X, Y] = -[Y, X]$ for any vector fields $X$ and $Y$
• Skew-symmetry implies that $[X, X] = 0$ for any vector field $X$, which is known as the alternating property
• Skew-symmetry is a fundamental property of Lie brackets and is closely related to the antisymmetry of the exterior product of differential forms

Bilinearity of Lie bracket

• The Lie bracket is bilinear, which means it is linear in each argument separately
• For vector fields $X$, $Y$, $Z$, and scalars $a$ and $b$, we have:
  • $[aX + bY, Z] = a[X, Z] + b[Y, Z]$
  • $[X, aY + bZ] = a[X, Y] + b[X, Z]$
• Bilinearity allows us to compute Lie brackets of linear combinations of vector fields by taking the corresponding linear combinations of their Lie brackets

Relation to commutator of derivations

• The Lie bracket of vector fields is closely related to the commutator of derivations on the algebra of smooth functions on a manifold
• For vector fields $X$ and $Y$, the Lie bracket $[X, Y]$ acts on smooth functions as the commutator of the derivations $X$ and $Y$, i.e., $[X, Y](f) = X(Y(f)) - Y(X(f))$
• This relation provides an algebraic interpretation of the Lie bracket and connects it to the concept of derivations on algebras

Relation between Lie derivative and Lie bracket

• The Lie derivative and Lie bracket are closely related concepts in differential geometry, and their relationship is captured by several important formulas and identities
• Understanding the connection between these two concepts is crucial for studying the geometry of manifolds and the behavior of tensor fields under the action of vector fields
• The most important relations between the Lie derivative and Lie bracket are the definition of the Lie derivative in terms of the Lie bracket and Cartan's magic formula

Lie derivative as Lie bracket with vector field

• The Lie derivative of a vector field $Y$ along another vector field $X$ can be defined as the Lie bracket of $X$ and $Y$, i.e., $\mathcal{L}_X Y = [X, Y]$
• This definition provides a direct link between the Lie derivative and the Lie bracket, and it allows us to interpret the Lie derivative as a measure of the non-commutativity of the flows generated by the vector fields $X$ and $Y$
• The Lie derivative of other tensor fields can also be expressed in terms of Lie brackets and the contraction operator, which generalizes the connection between the Lie derivative and Lie bracket

Cartan's magic formula

• Cartan's magic formula is an important identity that relates the Lie derivative, exterior derivative, and interior product of differential forms
• For a differential form $\omega$ and a vector field $X$, Cartan's magic formula states that $\mathcal{L}_X \omega = i_X d\omega + d(i_X \omega)$, where $i_X$ is the interior product with $X$ and $d$ is the exterior derivative
• This formula provides a powerful tool for computing Lie derivatives of differential forms and understanding their behavior under the action of vector fields
• Cartan's magic formula is a fundamental result in differential geometry and has numerous applications in mathematical physics, such as in the study of symplectic and contact manifolds

Applications of Lie derivative and Lie bracket

• The Lie derivative and Lie bracket have numerous applications in differential geometry, mathematical physics, and other areas of mathematics
• These concepts play a crucial role in the study of symmetries, conservation laws, and the integrability of distributions on manifolds
• Some of the most important applications of the Lie derivative and Lie bracket include the study of Frobenius theorem, the Lie algebra of vector fields on manifolds, and the geometry of Lie groups

Symmetries and conservation laws

• The Lie derivative and Lie bracket are essential tools in the study of symmetries and conservation laws in mathematical physics
• A vector field $X$ on a manifold $M$ is called a symmetry of a tensor field $T$ if $\mathcal{L}_X T = 0$, which means that $T$ is invariant under the flow generated by $X$
• Symmetries of a system of differential equations can be used to find conserved quantities and simplify the analysis of the system
• The Lie bracket of symmetry vector fields forms a Lie algebra, which captures the structure of the symmetry group of the system

Frobenius theorem and integrability

• The Frobenius theorem is a fundamental result in differential geometry that characterizes the integrability of distributions on manifolds
• A distribution $\Delta$ on a manifold $M$ is a smooth assignment of a subspace of the tangent space at each point of $M$
• The Frobenius theorem states that a distribution $\Delta$ is integrable (i.e., it corresponds to a foliation of $M$) if and only if it is involutive, which means that the Lie bracket of any two vector fields in $\Delta$ also belongs to $\Delta$
• The Lie bracket plays a crucial role in the formulation and proof of the Frobenius theorem, and it provides a powerful tool for studying the integrability of distributions and the existence of local coordinate systems adapted to a given distribution

Lie algebra of vector fields on manifolds

• The space of vector fields on a manifold $M$ has the structure of a Lie algebra under the Lie bracket operation
• This Lie algebra, denoted by $\mathfrak{X}(M)$, captures the infinitesimal symmetries of the manifold and plays a fundamental role in the study of the geometry and topology of $M$
• Many important geometric structures on manifolds, such as Riemannian metrics, symplectic forms, and Poisson brackets, can be studied using the Lie algebra of vector fields and its representations
• The Lie algebra of vector fields is also closely related to the concept of Lie groups and their actions on manifolds, providing a bridge between the infinitesimal and global aspects of symmetry in differential geometry