1.7 KiB
Fiber bundles
Let X
be a manifold over a field F
.
Definition 1: a fiber
V_x
at a pointx \in X
on a manifold is a finite dimensional vector space. With the collection of fibersV_x
for allx \in X
define the fiber bundle as
V = \bigcup_{x \in X} V_x.
Then by definition we have the projection map \pi
given by
\pi: V \to X: (x,\mathbf{v}) \mapsto \pi(x, \mathbf{v}) \overset{\text{def}}{=} x,
and its inverse
\pi^{-1}: X \to V: x \mapsto \pi(x) \overset{\text{def}}{=} V_x.
Similarly, a dual fiber V_x^*
may be defined for x \in X
, with its fiber bundle defined by
V^* = \bigcup_{x \in X} V_x^*.
Definition 2: a tensor fiber
\mathscr{B}_x
at a pointx \in X
on a manifold is defined as
\mathscr{B}x = \bigcup{p,q \in \mathbb{N}} \mathscr{T}^p_q(V_x).
With the collection of tensor fibers
\mathscr{B}_x
for allx \in X
define the tensor fiber bundle as
\mathscr{B} = \bigcup_{x \in X} \mathscr{B}_x.
Then for a point x \in X
we have a tensor \mathbf{T} \in \mathscr{B}_x
such that
\mathbf{T} = T^{ij}_k \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k,
with T^{ij}_k \in \mathbb{K}
holors of \mathbf{T}
. Furthermore, we have a basis \{\mathbf{e}_i\}_{i=1}^n
of V_x
and a basis \{\mathbf{\hat e}^i\}_{i=1}^n
of V_x^*
.
Definition 3: a tensor field
\mathbf{T}
on a manifoldX
is a section
\mathbf{T} \in \Gamma(X, \mathscr{B}),
of the tensor fiber bundle
\mathscr{B}
.
Therefore, a tensor field assigns a tensor fiber (or tensor) to each point on a section of the manifold. These tensors may vary smoothly along the section of the manifold.