Andreas Cap
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A conjecture about parallelizable generalized spheres
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14 votes

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 $S^d$ ...

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Characterization of the exterior derivative
12 votes

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 ...

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What is the relation between spherical principal series representations of a reductive Liegroup and Verma modules for its Lie algebra?
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10 votes

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 ...

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Different definitions of spin structures
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10 votes

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 ...

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Tangent bundle of a homogeneous space and the euler exact sequence
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8 votes

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 ...

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de Rham cohomology and flat vector bundles
7 votes

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 ...

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Do contact and CR structures have corresponding $G$-structures?
6 votes

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 ...

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An Stokes type theorem for some operations other than integral
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5 votes

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 ...

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Commuting of exterior derivative and contraction (vector-valued forms)
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5 votes

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 $\nabla^V$ ...

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Global section of universal bundle on Grassmanian
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5 votes

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 ...

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triviality of Whitney sums of a vector bundle
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5 votes

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 ...

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What is the space for the coefficients of the connection 1-form of a connection in a vector bundle?
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5 votes

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 ...

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On generalized Tanaka connection
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4 votes

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 ...

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Constant spinors from constant forms
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4 votes

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 $...

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The standard projective cotractor bundle and its cocycle of transition functions
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3 votes

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 $\...

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What is the space parametrising the curved sub-Cartan geometries of a flat Cartan geometry?
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3 votes

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 ...

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Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?
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3 votes

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 ...

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Isomorphism of Complex Stiefel manifold and Homogeneous space of unitary group, and the Stiefel logarithm problem
2 votes

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 ...

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What is the curved version of the Tits fibration for $G_2$?
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2 votes

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 ...

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Constructing jet bundles from a cocycle of smooth transition functions
2 votes

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 ...

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Geometric interpretation of splitting of sequence associated to a homogeneous space
2 votes

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 ...

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canonical action of symmetric groups on orthogonal groups
2 votes

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 ...

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Does a $G$-structure on $M$ automatically descend to a contact $G$-structure on $\mathbb{P}T^*M$?
1 votes

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\...

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Conjugate Matrix
1 votes

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 ...

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