6.2 Killing vector fields

Killing vector fields are essential in understanding symmetries in Riemannian geometry. These special vector fields preserve the metric tensor, generating isometries that leave the manifold's structure unchanged. They provide insights into the fundamental properties of geometric spaces.

Killing fields are characterized by Killing's equation, which relates to the Lie derivative of the metric tensor. By solving this equation, we can identify the symmetries of a manifold and understand its isometries, leading to important applications in physics and differential geometry.

Definition of Killing vector fields

• Killing vector fields are special vector fields on a Riemannian or pseudo-Riemannian manifold that preserve the metric tensor
• Intuitively, Killing fields generate isometries, which are transformations that leave the metric unchanged
• Formally, a vector field $X$ is a Killing field if the Lie derivative of the metric tensor $g$ with respect to $X$ vanishes: $\mathcal{L}_X g = 0$

Lie derivative of metric tensor

• The Lie derivative $\mathcal{L}_X g$ measures the change in the metric tensor $g$ along the flow generated by the vector field $X$
• It captures the infinitesimal change in the metric when the manifold is slightly deformed in the direction of $X$
• The Lie derivative can be expressed in terms of the covariant derivative: $(\mathcal{L}_X g)_{ij} = \nabla_i X_j + \nabla_j X_i$

Killing's equation

• Killing's equation is the fundamental equation that characterizes Killing vector fields
• In local coordinates, Killing's equation takes the form: $\nabla_i X_j + \nabla_j X_i = 0$
• This equation expresses the condition that the metric tensor remains unchanged along the flow of the Killing field

Coordinate expression of Killing's equation

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• In a coordinate basis, Killing's equation can be written using Christoffel symbols: $\partial_i X_j + \partial_j X_i - 2\Gamma^k_{ij} X_k = 0$
• This form of Killing's equation involves partial derivatives of the components of the Killing field and the Christoffel symbols of the metric
• Solving this system of partial differential equations allows us to find the Killing fields on a given manifold

Isometries and Killing fields

• Isometries are transformations that preserve the metric structure of a manifold
• Every Killing field generates a one-parameter group of isometries, and conversely, every one-parameter group of isometries has an associated Killing field
• The flow of a Killing field consists of isometries, which means that the metric remains unchanged along the integral curves of the Killing field

One-parameter groups of isometries

• A one-parameter group of isometries is a family of isometries $\phi_t$ parameterized by a real number $t$, satisfying certain properties:
  • $\phi_0$ is the identity transformation
  • $\phi_t \circ \phi_s = \phi_{t+s}$ for all $t,s \in \mathbb{R}$
  • The map $(t,p) \mapsto \phi_t(p)$ is smooth for all $p$ in the manifold
• The Killing field associated with a one-parameter group of isometries is obtained by differentiating the isometries with respect to the parameter $t$ at $t=0$

Orbits of Killing fields

• The orbit of a point $p$ under a Killing field $X$ is the set of all points that can be reached by following the integral curves of $X$ starting from $p$
• Orbits of Killing fields provide a way to understand the symmetries and structure of the manifold
• The dimension of the orbit (orbit type) can vary from point to point, depending on the nature of the Killing field

Fixed points of isometries

• A fixed point of an isometry $\phi$ is a point $p$ such that $\phi(p) = p$
• Fixed points of isometries correspond to zeros of the associated Killing field
• The behavior of the Killing field near a fixed point can provide information about the local geometry of the manifold

Algebra of Killing fields

• The set of all Killing fields on a manifold forms a Lie algebra under the Lie bracket operation
• The Lie bracket of two Killing fields is another Killing field, which means that Killing fields are closed under the Lie bracket
• The dimension of the Lie algebra of Killing fields is related to the symmetries and isometries of the manifold

Lie bracket of Killing fields

• The Lie bracket of two vector fields $X$ and $Y$ is defined as $[X,Y] = XY - YX$, where $XY$ denotes the directional derivative of $Y$ along $X$
• For Killing fields $X$ and $Y$, the Lie bracket $[X,Y]$ is also a Killing field
• The Lie bracket captures the non-commutativity of the flows generated by the Killing fields

Constant length of Killing fields

• Killing fields have constant length along their integral curves
• The inner product $\langle X, X \rangle$ of a Killing field $X$ with itself is constant along the flow of $X$
• This property follows from the fact that Killing fields preserve the metric tensor

Killing fields vs conformal Killing fields

• Conformal Killing fields are a generalization of Killing fields that preserve the metric tensor up to a scalar factor
• A vector field $X$ is a conformal Killing field if $\mathcal{L}_X g = \lambda g$ for some scalar function $\lambda$
• Killing fields are a special case of conformal Killing fields, where $\lambda = 0$

Examples of Killing fields

• Killing fields arise naturally in many important geometries and provide insights into their symmetries and properties
• Studying examples of Killing fields helps develop intuition and understanding of their behavior

Killing fields in Euclidean space

• In Euclidean space $\mathbb{R}^n$, the Killing fields correspond to translations and rotations
• Translation Killing fields are of the form $X = (a_1, \ldots, a_n)$, where $a_i$ are constants
• Rotation Killing fields are of the form $X = (-y, x, 0, \ldots, 0)$, $(0, -z, y, 0, \ldots, 0)$, etc., representing rotations in different planes

Killing fields on spheres

• On the unit sphere $S^n$, the Killing fields generate rotations around various axes
• The dimension of the space of Killing fields on $S^n$ is $\frac{1}{2}n(n+1)$
• For example, on the 2-sphere $S^2$, there are three linearly independent Killing fields corresponding to rotations around the $x$, $y$, and $z$ axes

Killing fields in hyperbolic space

• In hyperbolic space $\mathbb{H}^n$, the Killing fields correspond to hyperbolic translations and rotations
• The dimension of the space of Killing fields in $\mathbb{H}^n$ is $\frac{1}{2}n(n+1)$
• Hyperbolic translations move points along geodesics, while hyperbolic rotations preserve certain hypersurfaces

Physical significance of Killing fields

• Killing fields have important applications in physics, particularly in the context of general relativity and gauge theories
• They provide a mathematical framework for understanding symmetries and conservation laws in physical systems

Conservation laws from Killing fields

• In general relativity, Killing fields are closely related to conservation laws and conserved quantities
• If a spacetime admits a Killing field, then there exists a corresponding conserved quantity along the geodesics of the spacetime
• For example, time translation symmetry leads to conservation of energy, while rotational symmetry leads to conservation of angular momentum

Computing Killing fields

• Finding Killing fields on a given manifold involves solving Killing's equation, which is a system of partial differential equations
• The process of computing Killing fields depends on the specific geometry and coordinates used

Solving Killing's equation

• To find Killing fields, one needs to solve Killing's equation $\nabla_i X_j + \nabla_j X_i = 0$ for the components of the vector field $X$
• In practice, this often involves using the coordinate expression of Killing's equation and solving the resulting system of PDEs
• Symmetry considerations and the properties of the manifold can simplify the process of solving Killing's equation

Dimension of space of Killing fields

• The maximum number of linearly independent Killing fields on a manifold is determined by its dimension and geometric properties
• For an $n$-dimensional Riemannian manifold, the maximum dimension of the space of Killing fields is $\frac{1}{2}n(n+1)$
• However, not all manifolds admit the maximum number of Killing fields, and the actual dimension depends on the specific geometry