a function assign a tensor to each point of a region of a mathematical space (typically a Euclidean space or a manifold)
Definition
Let be a manifold, for instance the Euclidean plane
Then a tensor field of type is a section
where is a vector bundle on , is its dual and is the tensor product of vector bundles
See also (McConnell, 2014; Schouten, 1951)
a few math definitions
A tangent space is a -dimensional differentiable manifold associated with each point .
a non-degenerate, smooth, symmetric bilinear map that assigns a real number to pairs of tangent vectors at each tangent space of the manifold.
metric tensor
The map is symmetric and bilinear, so if are tangent vectors at point to the manifold then we have:
for any real number
is non-degenerate means there is no non-zero such that
a topological construction that makes precise the idea of of a family of vector space parameterised by another space
ex: could be a topological space, a manifold
Möbius strip
to every point of the space we “attach” a vector space in such a way that these vector space fits together to form another space of the same kind as
definition
A real vector bundle consists of
- topological spaces (base space) and (total space)
- a continuous surjection (bundle projection)
- For every in the structure of a finite-dimensional real vector space on the fiber
compatibility condition
For every point in , there is an open neighborhood of and a homeomorphism
such that for all in :
- for all vectors in
- the map is a linear isomorphism between vector spaces and
- open neighborhood together with the hoemomorphism is called a local trivialisation of the vector bundle 1
-
every fiber is a finite-dimensional real vector space and hence has a dimension
-
function is locally constant, and therefore constant on each connected component of
rank of the vector bundle
if is equal to constant on all of , then is the rank of the vector bundle, and is a vector bundle of rank
trivial bundle
The Cartesian product equipped with the projection is considered as the trivial bundle of rank over
operations on vector bundle extending the operation of duality for vector space.
definition
a dual bundle of a vector bundle is the vector bundle whose fiber are the dual spaces to fibers of
Equivalently, can be defined as the Hom bundle , the vector bundle of morphisms from to the trivial line bundle
a space that is locally a product space, but globally may have different topological structure
definition
A fiber bundle is a structure where:
- are topological space
- is a continuous surjection satisfying local triviality condition
is considered as base space, is total space, and is the fiber space
the map is called the projection map
consequences
we require that for every , there is an open neighborhood of such that there is a homeomorphism such that a way agrees with the projection onto the first factor. 2
where is the natural projection and is a homeomorphism.
The set of all is called a local trivialization of the bundle
Therefore, for any , the preimage is homeomorphic to 3 and is called the fiber over p
annotation
a fiber bundle is often denoted as
Suppose that and are base space, and and are fiber bundles over and respectively.
definition
bundle map/morphism consists of a pair of continuous functions
such that . That is the following is commutative:
Bibliographie
- McConnell, A. J. (2014). Applications of Tensor Analysis. Dover Publications. https://books.google.ca/books?id=ZCP0AwAAQBAJ
- Schouten, J. A. (1951). Tensor Analysis for Physicists. Oxford University Press.