It is true in general that $E_d$ is trivial. As remarked by Neil Strickland above, this boils down to showing that $TS^d\oplus\Lambda^2 TS^d$ is always a trivial bundle. To see this, represent $S^d$ ...

Here is a sketch of one possibility how to prove this: The first key step is to see that the operator you are looking at has to be a differential operator. One usual way to ensure this is to require ...

I don't know whether this helps, but there is a nice description of the relation between the two modules in geometric terms. The principal series representation can be viewed as the space of smooth ...

As it stands, the second definition is a concrete description of the spin group in dimension four. It defines an action of the simply connected group $SU(2)\times SU(2)$ on a vector space of real ...

The isomorphism $G\times_H(\mathfrak g/\mathfrak h)\to T(G/H)$ is induced by the map $G\times (\mathfrak g/\mathfrak h)$ mapping $(g,X+\mathfrak h)$ to $T_gp\cdot L_X(g)\in T_{gH}(G/H)$. The Euler ...

There is a another setting in which this is very useful, namely when the flat vector bundle comes from a locally flat geometric structure on a manifold. The best known example is the one of a sphere ...

I think that the relation of contact structures and CR structures to classical G-structures is a bit more complicated than suggested by Ben McKay's reply, although I certainly agree with his comments ...

If you assume that $M$ is oriented, then up to a multiple $I_1$ and $I_2$ are the usual integral. In this case $\partial M$ is oriented, and since this is a manifold without boundary, the integral ...

The general version you are looking for is the following: Suppose that$\Phi:V\to W$ is a vector bundle map between two vector bundles $V$ and $W$ over $M$. Suppose that we have connections $\nabla^V$ ...

These are simple instances of the Bott-Borel-Weil theorem. For a complex semsimple group $G$ and a parabolic subgroup $P$ and a complex irreducible representation $W$ of $P$ consider the homogeneous ...

There is a simple general argument showing that $\xi$ is trivial: Take any homogeneous space $G/H$ and any representation $V$ of $G$ and restrict the representation to $H$. Then the homogeneous vector ...

This answer just extends the remark by Liviu Nicolaescu above. For a general connection on a vector bundle with structure group $G$, you cannot say anything about the connection coefficients. (As an ...

The short answer to your question is that the Levi-Civita connection is perfectly adapted to the metric, but not compatible with the additional structure around. In particular, it does not preserve ...

If you also assume that $\nabla$ preserves the complex structure, it is true that $\nabla^{S_{\mathbb C}}\eta=0$ if and only if $\nabla\Omega=0$. The assumptions mean that the unitary frame bundle of $...

There are several aspects to your question, and I think that asking for a cocycle of transition functions is partly misleading. The point here is that as you observe in the quesiton, you can define $\...

I don't think that this is a real answer to your question, but there is a general concept of extension functors for Cartan geometries. Such a functor extending Cartan geometries of type $(G,P)$ to ...

I also believe that the answer to your question is yes, but I think that things are more complicated than indicated in the answer by @Ben_McKay and in your partial solution. The point is that while ...

The map $U(n)\to V_k(\mathbb C^n)$ simply maps a matrix $U$ to its first $k$ columns, which are $k$ orthonormal vectors in $\mathbb C^n$. Otherwise put, $U$ is mapped to $(Ue_1,\dots,Ue_k)$ and the ...

This is an instance of the general construction of correspondence spaces and twistor spaces as described in my article MR2139714 and in Chapter 4 of the book of Jan Slovak and myself on parabolic ...

I would not really call this an answer but rather an extended comment. I think that the answer by @IgorKhavkine is correct but at the same time misleading in a certain sense. The point about this is ...

I am not aware of a general "geometric interpretation" of such a vector space splitting in the non-equivariant case, but there is at least one interesting application, which is along the lines of the ...

I think the bundle you consider is trivial, because the representation of $S_{n+1}$ on $\mathbb R^{n+1}$ by permutation of coordinates extends to a representation of $O(n)$. I am not completely sure ...

The $CSp(2n)$-structure you describe makes sense and is an equivalent encoding of the contact structure on the projectivized cotangent bundle. Let us start with a general contact manifold $N$, $H\...

I don't think that there can be a positive answer to either of the two questions. The matrix $J$ is equivalent to a fixed decomposition of your intial vector space into two linear subspaces of the ...