
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a frame bundle is a
principal fiber bundle associated with any
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 (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
''
''. The fiber of
over a point ''
'' is the set of all
ordered bases, or ''frames'', for ''
''. The
general linear group
In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
acts naturally on
via a
change of basis, giving the frame bundle the structure of a principal ''
''-bundle (where ''k'' is the rank of ''
'').
The frame bundle of a
smooth manifold is the one associated with its
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
. For this reason it is sometimes called the tangent frame bundle.
Definition and construction
Let ''
'' be a real
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 (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
of rank ''
'' over a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''
''. A frame at a point ''
'' is an
ordered basis for the vector space ''
''. Equivalently, a frame can be viewed as a
linear isomorphism
:
The set of all frames at ''
'', denoted ''
'', has a natural
right action by the
general linear group
In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
''
'' of invertible ''
'' matrices: a group element ''
'' acts on the frame ''
'' via
composition to give a new frame
:
This action of ''
'' on ''
'' is both
free and
transitive (this follows from the standard linear algebra result that there is a unique invertible linear transformation sending one basis onto another). As a topological space, ''
'' is
homeomorphic to ''
'' although it lacks a group structure, since there is no "preferred frame". The space ''
'' is said to be a ''
''-
torsor.
The frame bundle of ''
'', denoted by
or
, is the
disjoint union of all the ''
'':
:
Each point in
is a pair (''x'', ''p'') where ''
'' is a point in ''
'' and ''
'' is a frame at ''
''. There is a natural projection
which sends ''
'' to ''
''. The group ''
'' acts on
on the right as above. This action is clearly free and the
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
s are just the fibers of ''
''.
Principal bundle structure
The frame bundle
can be given a natural topology and bundle structure determined by that of ''
''. Let ''
'' be a
local trivialization of ''
''. Then for each ''x'' ∈ ''U''
''i'' one has a linear isomorphism ''
''. This data determines a bijection
:
given by
:
With these bijections, each ''
'' can be given the topology of ''
''. The topology on
is the
final topology coinduced by the inclusion maps ''
''.
With all of the above data the frame bundle
becomes a
principal fiber bundle over ''
'' with
structure group ''
'' and local trivializations ''
''. One can check that the
transition functions of
are the same as those of ''
''.
The above all works in the smooth category as well: if ''
'' is a smooth vector bundle over a
smooth manifold ''
'' then the frame bundle of ''
'' can be given the structure of a smooth principal bundle over ''
''.
Associated vector bundles
A vector bundle ''
'' and its frame bundle
are
associated bundles. Each one determines the other. The frame bundle
can be constructed from ''
'' as above, or more abstractly using the
fiber bundle construction theorem. With the latter method,
is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as ''
'' but with abstract fiber ''
'', where the action of structure group ''
'' on the fiber ''
'' is that of left multiplication.
Given any
linear representation ''
'' there is a vector bundle
:
associated with
which is given by product
modulo the
equivalence relation ''
'' for all ''
'' in ''
''. Denote the equivalence classes by ''