Topological notions

Fiber bundles

Let \(X\) be a manifold over a field \(F\).

Definition 1: a fiber \(V_x\) at a point \(x \in X\) on a manifold is a finite dimensional vector space. With the collection of fibers \(V_x\) for all \(x \in X\) 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 point \(x \in X\) 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 all \(x \in X\) 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 manifold \(X\) 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.