notes/docs/mathematics/differential-geometry/topological-notions.md

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.