Tensor field

A tangent space is a $n$-dimensional differentiable manifold $M$ associated with each point $p$.

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 $g$

$$g: T_p M \times T_p M \to \mathbb{R}$$

The map is symmetric and bilinear, so if $X, Y, Z \in T_p M$ are tangent vectors at point $p$ to the manifold $M$ then we have:

for any real number $a \in \mathbb{R}$

$g$ is non-degenerate means there is no non-zero $X \in T_p M$ such that $g(X,Y)=0 \forall \space Y \in T_p M$

a topological construction that makes precise the idea of of a family of vector space parameterised by another space $X$

ex: $X$ could be a topological space, a manifold

Möbius strip

to every point $x$ of the space $X$ we “attach” a vector space $V(x)$ in such a way that these vector space fits together to form another space of the same kind as $X$

definition

A real vector bundle consists of

• topological spaces $X$ (base space) and $E$ (total space)
• a continuous surjection $\pi: E \rightarrow X$ (bundle projection)
• For every $x$ in $X$ the structure of a finite-dimensional real vector space on the fiber $\pi^{-1}(\{x\})$

compatibility condition

For every point $p$ in $X$, there is an open neighborhood $U \subseteq X$ of $p$ and a homeomorphism

$$\varphi : U \times \mathbb{R}^k \rightarrow \pi^{-1}(U)$$

such that for all $x$ in $U$:

• $(\pi \circ \varphi)(x,v)=x$ for all vectors $v$ in $\mathbb{R}^k$
• the map $v \mapsto \varphi(x,v)$ is a linear isomorphism between vector spaces $\mathbb{R}^k$ and $\pi^{-1}(\{x\})$

operations on vector bundle extending the operation of duality for vector space.

definition

a dual bundle of a vector bundle $\pi : E \rightarrow X$ is the vector bundle $\pi^{*}: E^{*} \rightarrow X$ whose fiber are the dual spaces to fibers of $E$

Equivalently, $E^{*}$ can be defined as the Hom bundle $\text{Hom}(E, \mathbb{R} \times X)$, the vector bundle of morphisms from $E$ to the trivial line bundle $\mathbb{R} \times X \rightarrow X$

a space that is locally a product space, but globally may have different topological structure

definition

A fiber bundle is a structure $(E, B, \pi, F)$ where:

• $E, B, F$ are topological space
• $\pi: E \rightarrow B$ is a continuous surjection satisfying local triviality condition

$B$ is considered as base space, $E$ is total space, and $F$ is the fiber space

the map $\pi$ is called the projection map

consequences

we require that for every $x \in B$, there is an open neighborhood $U \subseteq B$ of $x$ such that there is a homeomorphism $\varphi: \pi^{-1}(U) \rightarrow U \times F$ such that a way $\pi$ agrees with the projection onto the first factor. ²

where $\text{proj}_1: U \times F \rightarrow U$ is the natural projection and $\varphi : \pi^{-1}(U) \rightarrow U \times F$ is a homeomorphism.

The set of all $\{(U_i, \varphi_i)\}$ is called a local trivialization of the bundle

Therefore, for any $p \in B$, the preimage $\pi^{-1}(\{p\})$ is homeomorphic to $F$ ³ and is called the fiber over p

annotation

a fiber bundle $(E, B, \pi, F)$ is often denoted as

$$F \to E \xrightarrow{\pi} B$$