Motivation of vector bundle of a manifold I am studying about vector bundle from M.Lee but not getting the feel of it. Can someone explain me about the importance of vector bundle? Why do we need to study about it?
Thanks!
Also, i have to study about connection of vector bundle so i wanted to know about the  motivation of connection of vector bundle also. If someone can tell me about its importance and motivation then it would be great!
Thanks again!
 A: Manifolds are spaces that locally look like $\mathbb{R}^n$. Similarly, vector bundles are spaces that locally look like $\mathbb{R}^n \times V$, where $V$ is some vector space. The simplest kind of vector bundle is a trivial bundle $M \times V$, if $M$ is a manifold, but the need for nontrivial vector bundles is seen immediately from looking at the tangent space.
A section of the bundle $\mathbb{R}^n \times V$ is just a smooth function $\phi:\mathbb{R}^n \rightarrow V$. If you have such a function you can, for instance, take partial derivatives: $\frac{\partial \phi}{\partial x_i}$, and this lets you talk about differential equations. However, it's not so easy on a nontrivial vector bundle: The reason is that if $x$ and $y$ are two points close together on a manifold you can't take $\phi(x) - \phi(y)$- they lie in different vector spaces. A connection is a gadget that allows you to take partial derivatives and formulate PDE's/field theories on a manifold.
A: The aim of differential geometry is to do calculus on more general mathematical objects than just the standard vector space $\mathbb{R}^{n}$. To be able to consider directional derivatives on a manifold $M$ one is led to the notion of a tangent space: the vector space of all directional derivative operators at a given point. The disjoint union of all the tangent spaces of the manifold yields the notion of the tangent bundle of $M$‚ denoted by $TM$. Now how is this related to vector bundles? The tangent bundle is a specific example of a vector bundle, simple as that. So in order to understand the tangent bundle, it is convenient to study vector bundles in general, and then in the context given, we restrict to tangent bundles alone.
And what about a connection? Think about the vector bundle as the tangent bundle for a while. A vector field is a smooth section of this bundle. In other words, at each point of the manifold we choose an element of the tangent space above that point, and we require that this decision varies smoothly from point to point. It is then interesting to ask, that when given two vector fields $X$ and $Y$, how does the vectors of $Y$ vary in the direction of the vectors given by $X$? In other words, we want to define a way to differentiate vector fields with respect to each other. This is given by a notion of a connection on the tangent bundle. Where does the name connection come from? Here's an intuitive idea to think about it. One can show that given a connection on the bundle and a smooth path on the manifold, there exists a unique parallel transport operation that induces an isomorphism between the tangent spaces of the points on that path. In other words, "it connects" the tangent spaces on the direction of that path. These ideas will come clear once you cover the following topics in Lee's book: vector fields along curves, the covariant derivative operator $D_{t}$, and parallel transport.
