port from mathematics-physics notes

docs/mathematics/topology/fiber-bundles.md

@@ -0,0 +1,57 @@
+# 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.