^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] <<[[Schutz - 2.8 - Basis Vectors and Basis Vector Fields|2.8 Basis Vectors and Basis Vector Fields]] | [[Schutz - 2.10 - Examples of Fiber Bundles|2.10 Examples of Fiber Bundles]]>> # Fiber Bundles A particularly interesting [[Schutz - 2.1 - Definition of a Manifold#^31cc42|manifold]] is formed by combining a manifold $M$ with all its [[Schutz - 2.7 - Vectors and Vector Fields#^4e794a|tangent space]] $T_P\, \forall P \in M$. An example of this is shown in the figure below for a one-dimensional manifold. We draw the curve and a few tangent spaces, but since they are going to arbitrarily intersect and muddy the picture, we orient them parallel to each other instead as shown in subfigure (b). This gives the name fiber bundle, since the lines look like fibers. They only cross the manifold $M$ at points where they are defined. This specific manifold is known as the *tangent bundle*, $TM$, and we will define it more precisely below. Each *fiber* (i.e. tangent space) extends infinitely far in each direction, and each point on the fiber gives a vector whose "length" is equal. Each point of the in the figure (the combined manifold with its tangent spaces) is a point of one and only one tangent space. ![[Pasted image 20210126114753.png]] Let the one-dimensional manifold have coordinate $x$, and we want to find the coordinates of the tangent spaces to points of $M$ in the region $a<x<b$, for some $a, b \in \mathbb{R}$. We assume that the coordinate $x$ is a good coordinate in this interval. Any tangent vector $\pmb{V}$ to a point $P \in M$ can be written as: $ \pmb{V} = y \frac{\partial}{\partial x} $ so that $y$ is a coordinate for $T_P$, a good coordinate over the whole fiber $T_P$. Since each fiber is defined on a single point in the manifold $x$, $(x,y) \in TM$ forms the *natural coordinate system* for the fiber bundle. Since every point in the fiber bundle lies in such a region, then the fiber bundle is a manifold. A curve in the fiver bundle as shown in the figure below (dashed) line identifies a particular vector with each point of the manifold $M$. Thus, it is a vector field, called a cross-section of $TM$. > **Fiber Bundle** > A *fiber bundle* consists of a base manifold $M$ together with a *fiber* $F_P$ attached to each point $P \in M$ of the base manifold. If the base space is $n$ dimensional and each fiber is $m$ dimensional, then the fiber bundle is an $m+n$ dimensional manifold. It is equipped with a projection map $\pi:F\rightarrow M$ which maps to each point of a fiber a point in the base manifold. A more precise definition is given in section 2.11 in the book. > **Tangent Bundle** > The *tangent bundle* is a $2n$ dimensional fiber bundle whose base manifold is an n-dimensional manifold $M$ and the fibers are the tangent spaces at every point $P \in M$. It can be written as: > $ TM = \bigcup_{x \in M} \{(x, y), y \in T_x M\} $ > This is an example of a *vector bundle*, a fiber bundle whose fibers are vector spaces. ^7a66b2