# Discussion and Definition > **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 particularly interesting [[manifold]] is formed by combining a manifold $M$ with all its [[tangent spaces]] $T_P\, \forall P \in M$. An example of this is shown in the figure below for a one-dimensional [[manifold]]. We draw the curve and a few [[Tangent Space|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]], $TM$. 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 [[Coordinate Basis|good coordinate]] in this interval. Any [[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 fiber 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 $TM$*. # Example Newtonian mechanics can be formulated on a fiber bundle, whose base manifold is time $\mathbb{R}^1$ and whose fibers are Euclidean 3-space $\mathbb{R}^3$. This is because time is absolute: everyone can agree what events are simultaneous no matter where they occur in the fiber $\mathbb{R}^3$. There is no natural relation between points at different fibers (i.e. at different times). ![[Pasted image 20210126121230.png]]