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If $X$ is a manifold, $G$ a compact Lie group, $E$ a principal $G$-bundle over $X$ and $V$ be a vector space on which $G$ acts. Then one can form the vector bundle $E \times_{G} V$ over $X$. What is the intuition for this bundle? Do you know any good picture to have in mind, how this bundle looks like?

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I've always comforted myself with the slogan that «the induced bundle is the result of glueing copies of the fiber $F$ twisted in the same way as those of $E$». –  Mariano Suárez-Alvarez Jan 4 '12 at 4:01
@Mariano: Dear Mariano, That's a nicely succinct way of saying what I tried to say in my answer! Best wishes, –  Matt E Jan 4 '12 at 6:00
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up vote 4 down vote accepted

To give the principal $G$-bundle $E$ is the same as giving a cover of $X$ by balls $U_i$, and a $G$-valued smooth function $\varphi_{ij}$ on each intersection $U_i \cap U_j$, satisfying an appropriate cocyle condition for any three indices $i, j,$ and $k$. (The bundle $E$ is then obtained by gluing the trivial bundles $U_i \times G$ together using right translation by $\varphi_{ij}$ to identify the copy of $(U_i \cap U_j) \times G$ inside $U_i \times G$ with the copy of $(U_i \cap U_j) \times G$ inside $U_j \times G$.)

In short, $E$ is determined by the transition functions $\varphi_{ij}$.

Now if we are given a representation $\rho\colon G\to GL_n(V)$, we can apply it to the transition functions $\varphi_{ij}$, to get a collection of $GL(V)$-valued transition functions $\rho\circ\varphi_{ij}$. Using these, we can glue the various trivial bundles $U_i \times V$ together to obtain the induced bundle $E\times_G V$.

To summarize: a principal bundle is given by a bunch of transition functions, taking values in $G$ (some Lie group). A representation $\rho$ of $G$ allows us to convert these into $GL_n$ valued transition functions (if $V$ has dimension $n$), which determine a vector bundle.

Of course, all I am doing is recalling the construction from a Cech point of view, but this is a good way to think about it, in my view.

Another way to think about it is as follows: if $\psi\colon G \to H$ is any homomorphism of Lie groups, then given the transition functions $\varphi_{ij}$ for the principal bundle $E$, we may compose them with the homomorphism $\psi$ to obtain transition functions $\psi\circ \varphi_{ij}$ for a principal $H$-bundle (again referred to as an induced bundle, which can be written as $E\times_G H$).

Now if $\rho:G \to GL(V)$ is a linear representation of $G$, we can apply this process to obtain a principal $GL(V)$ bundle (i.e. a principal $GL_n$ bundle, if $V$ has dimension $n$).

Your question then reduces to the following: why are principal $GL_n$ bundles the same thing as $n$-dimensional vector bundles. The answer is: given the principal $GL_n$-bundle, using the standard $n$-dimensional rep'n of $GL_n$, we can obtain an $n$-dimensional vector bundle via the induced bundle procedure. Conversely, given an $n$-dimensional vector bundle, we can form the associated frame bundle, which is a principal $GL_n$-bundle. These two processes are (quasi-)inverse to one another.

So, if you are comfortable with moving between principal $GL_n$-bundles and $n$-dimensional vector bundles, then you can think of forming the induced bundle as just replacing the structure group $G$ with $GL(V)$ via the representation $\rho$. If you aren't comfortable with interchanging principal $GL_n$-bundles and $n$-dimensional vector bundles, then you should practice with those ideas until you do become comfortable, but they are what underly the motivations and constructions in the general theory of principal bundles.

Now here are some examples you can practice with. Here is one:

given an $m$-dim'l vector bundle $V$ and an $n$-dim'l vector bundle $W$, convert them into principle bundles $E$ and $F$ for the groups $GL_m$ and $GL_n$. Now take the fibre product $E\times_X F$; this is a principle bundle for the group $GL_m \times GL_n$. Now $GL_m \times GL_n$ has a natural representation $\rho$ into $GL_{m+n}$, given by taking the direct sum of two matrices.

Exercise: Show that the bundle induced by the homomorphism $\rho$ is precisely the direct sum of $V$ and $W$.

Exercise: Find the analogous description for the tensor product of $V$ and $W$.

Exercise: Compare with the examples in Adam Smith's answer.

In general, as these exercises suggest, you can think of induced bundles as a kind of generalization of the usual multilinear algebra of vector bundles.

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I think the key examples to have in mind are the following. I'm going to forget about $G$ being compact, since I don't think that's the important point. Associated to the tangent bundle $TX$ of $X$ is a principal $GL_n(\mathbb{R})$-bundle $E$. We then have the following standard examples.

1) Of course, the bundle induced from the usual action of $GL_n(\mathbb{R})$ on $\mathbb{R}^n$ is the tangent bundle $TX$.

2) The group $GL_n(\mathbb{R})$ also acts on the dual space $(\mathbb{R}^n)^{\ast}$ via $(M \cdot \phi)(\vec{v}) = \phi(M^{-1} \cdot \vec{v})$, where $\phi \in (\mathbb{R}^n)^{\ast}$ and $\vec{v} \in \mathbb{R}^n$. The bundle induced by that action is the cotangent bundle.

3) The group $GL_n(\mathbb{R})$ also acts on $\wedge^k (\mathbb{R}^n)^{\ast}$ in the usual way. The bundle induced by that action is the bundle of differential $k$-forms.

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