Killing vector fields are essential in understanding in Riemannian geometry. These special vector fields preserve the , 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 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- 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: LXg=0
Lie derivative of metric tensor
The Lie derivative LXg 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: (LXg)ij=∇iXj+∇jXi
Killing's equation
Killing's equation is the fundamental equation that characterizes Killing vector fields
In local coordinates, Killing's equation takes the form: ∇iXj+∇jXi=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: ∂iXj+∂jXi−2ΓijkXk=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 ϕt parameterized by a real number t, satisfying certain properties:
ϕ0 is the identity transformation
ϕt∘ϕs=ϕt+s for all t,s∈R
The map (t,p)↦ϕt(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 ϕ is a point p such that ϕ(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 ⟨X,X⟩ 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 LXg=λg for some scalar function λ
Killing fields are a special case of conformal Killing fields, where λ=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 Rn, the Killing fields correspond to translations and rotations
Translation Killing fields are of the form X=(a1,…,an), where ai are constants
Rotation Killing fields are of the form X=(−y,x,0,…,0), (0,−z,y,0,…,0), etc., representing rotations in different planes
Killing fields on spheres
On the unit sphere Sn, the Killing fields generate rotations around various axes
The dimension of the space of Killing fields on Sn is 21n(n+1)
For example, on the 2-sphere S2, 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 Hn, the Killing fields correspond to hyperbolic translations and rotations
The dimension of the space of Killing fields in Hn is 21n(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 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 ∇iXj+∇jXi=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 21n(n+1)
However, not all manifolds admit the maximum number of Killing fields, and the actual dimension depends on the specific geometry