Kosmann lift - AbsoluteAstronomy.com

In differential geometry, the Kosmann lift, named after Yvette Kosmann-Schwarzbach, of a vector field

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

is the canonical projection

on the orthonormal frame bundle of its natural lift

defined on the bundle of linear frames.

Generalisations exist for any given reductive G-structure

G-structure

In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....

Introduction

In general, given a subbundle

Subbundle

In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right....

of a fiber bundle

Fiber bundle

In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

over

and a vector field

, its restriction

is a vector field "along"

not on (i.e., tangent to)

. If one denotes by

the canonical embedding

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

, then

is a section

Section

Section may refer to:* Section * Section * Archaeological section* Histological section, a thin slice of tissue used for microscopic examination* Section, an instrumental group within an orchestra...

of the pullback bundle

Pullback bundle

In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...

, where

and

is the tangent bundle

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

of the fiber bundle

.
Let us assume that we are given a Kosmann decomposition of the pullback bundle

, such that

i.e., at each

one has

where

is a vector subspace of

and we assume

to be a vector bundle

Vector bundle

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 : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

over

, called the transversal bundle of the Kosmann decomposition. It follows that the restriction

splits into a tangent vector field

and a transverse vector field

being a section of the vector bundle

Definition

Let

be the oriented orthonormal frame bundle of an oriented

-dimensional
Riemannian manifold

with given metric

. This is a principal

-subbundle of

, the tangent frame bundle of linear frames over

with structure group

.
By definition, one may say that we are given with a classical reductive

-structure. The special orthogonal group

is a reductive Lie subgroup of

. In fact, there exists a direct sum

Direct sum

In mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...

decomposition

, where

is the Lie algebra of

, and

is the

-invariant vector subspace of symmetric matrices, i.e.

for all

Let

be the canonical embedding

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

.

One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle

Pullback bundle

such that

i.e., at each

one has

being the fiber over

of the subbundle

Subbundle

In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right....

. Here,

is the vertical subbundle of

and at each

the fiber

is isomorphic to the vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

of symmetric matrices

.

From the above canonical and equivariant

Equivariant

In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. Specifically, let G be a group and let X and Y be two associated G-sets. A function f : X → Y is said to be equivariant iffor all g ∈ G and all x in X...

decomposition, it follows that the restriction

of an

-invariant vector field

splits into a

-invariant vector field

, called the Kosmann vector field associated with

, and a transverse vector field

.

In particular, for a generic vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...