Questions tagged [vector-bundles]

For questions on vector bundles, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$.

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Specific homotopy between complex conjugation and the identity.

Consider the set $\mathcal{C} = C^{\infty}(\mathbb{C}^*, \mathbb{C}^*)$, where $\mathbb{C}^* = \mathbb{C}\backslash\{0\}$. Both $f(z) = z$ and $g(z) = \bar{z}$ can be seen as elements in $\mathcal{C}$...
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Is the trivial bundle ample on an affine variety?

When I was reading a paper, I came across a statement like "Since ince $M$ is affine, the trivial bundle is ample and ..." I think that line bundle $L$ on a variety $M$ is ample if it the global ...
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Why a section of $\otimes_{k}E^{*}$ defines a $k$-linear, fiber preserving map from $\oplus_{k}E$ to $M\times \mathbb{R}$?

I am not sure if this is a duplicate. Clifford Taubes assert in his book Differential Geometry that we may view sections of vector bundles as homomorphisms from $M\times \mathbb{R}$ to $E$ such that ...
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Trivilisations of Vector Bundles

Let $\pi : E \to X$ be a smooth rank $k$ vector bundle on a manifold $X$ (I don't think my question depends upon the stipulations on the bundle, but I've just chosen smooth in case I'm incorrect). By ...
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Classify the vector bundles of a manifold.

I met a question asking me to classify the $2$-dimensional vector bundles of the sphere $S^2$. I did not know how to classify the vector bundles in general. The only example I know was the line ...
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Does an orientable subbundle of an orientable vector bundle always have a orientable complement?

If I have an orientable vector bundle $E$ and a subbundle $F$ on a manifold $M$, where both the bundles are orientable, does $F$ have a complement in $E$ which is also orientable? Does it have a ...
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Orthogonal complement of a vector bundle

Let $E \rightarrow X$ be a vector bundle with an inner product. If $F$ is a sub-bundle, we can define an orthogonal complement bundle $F^\perp$ (see http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf ...
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Understanding the Definition of a Differential Form of Degree $k$

Let $M$ be a smooth manifold. A differential form of degree $k$ is a smooth section of the $k$th exterior power of the cotangent bundle of $M$. Does it mean that a differential form of degree $k$ ...
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Failure of isomorphisms on stalks to arise from an isomorphism of sheaves

It is well-known (Hartshorne 2.1.1) that if $F$ and $G$ are sheaves on a space $X$, then $\phi:F\rightarrow G$ is an isomorphism if and only if the induced stalk map $\phi_p:F_p\rightarrow G_p$ is an ...
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