For questions on vector bundles.

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Isomorphisms of bundles

Studying on vector bundles came across the following problem. Question: Let $M_1$ and $M_2$ Riemannian manifolds with $TM_1^{\perp}$ and $TM_2^{\perp}$ their normal of ranks $k_{1}$ and $k_{2}$, ...
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When does a principal divisor have degree 0?

We know that on a smooth curve over an algebraically closed field principal divisors have degree 0. This holds true also for the projective space over any field (in this case equality holds too). My ...
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Preimage of a submanifold is a submanifold - Transversality

It is well known that if a smooth Map $f : M \to N$ between two smooth manifolds (finite dimensional) is transversal to a submanifold $L \subset N, L \pitchfork f$, than $f^{-1}(N)$ is a submanifold ...
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Degree of vector bundle under pushforward while blowing up

Let $f:X\longrightarrow Y$ be a birational morphism of projective varieties over $\mathbb{C}$. In particular we can assume that $X$ is a blow of $Y$ at finitely many points. Let $F$ be a vector bundle ...
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Line bundles on $\mathbb{P}^2 \times \mathbb{P}^2$

I'm curious as to what are all the line bundles on $\mathbb{P}^2 \times \mathbb{P}^2$? I might be mistaken but I believe they are classified as $$ \mathcal{O}_{\mathbb{P}^2 \times \mathbb{P}^2}(p,q) ...
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Is there a natural Riemannian structure on the total space of a vector bundle?

Suppose $B$ is a Riemannian manifold and $\pi: E \to B$ is a smooth vector bundle equipped with a metric. Is there a natural Riemannian metric on $E$, i.e. a bundle metric on $TE\to E$? It seems ...
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Are these “Stiefel-Whitney numbers” invariants of the (smooth)topology of the total space?

Let $E$ be a smooth real vector bundle over a smooth compact $n$-dimensional manifold $M$. It is relatively easy to see that the Stiefel-Whitney classes $w_i(E)$ are not diffeomorphism invariants of ...
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Mobius over the sphere is the sphere itself

The Mobius band can be thought as a line bundle over $S^1$ by giving the vector spaces half a twist at some point. Now, we can do the same kind of construction by considering a line bundle over the ...
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Subsheaves of locally free sheaves on a rational curve

Let $k$ be an algebraically closed field, $\mathcal{E}$ a locally free sheaf on $\mathbb{P}^1_k$ and $\mathcal{L} \subset \mathcal{E}$ a subsheaf of rank $1$. By Grothendieck's theorem, we know that ...
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Is paralellizability a topological invariant (Invariant under homemorphism)

This MO post is a motivation to ask: Is paralellizability a topological invariant (Invariant under homeomorphism)?
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Vector bundle morphism as section of a bundle?

Let $\xi:=(E, p, M)$ and $\eta:=(F, q, N)$ be two smooth real vector bundles. A vector bundle morphism from $\xi$ to $\eta$ is a pair of smooth maps $(f:E\longrightarrow F, g:M\longrightarrow N)$ ...
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Two simple counterexamples in algebraic geometry

Suppose we have a smooth complex algebraic variety $X$. Then in general, $K^a(X)\to K(X^{an})$ is not surjective. Could someone give an example of a topological vector bundle class which contains no ...
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Vector bundles on Hirzebruch surface $\mathbb{F}_2$

I would like to know a classification for all holomorphic vector bundles on the second Hirzebruch surface $\mathbb{F}_2$. Is this known? What is known? In particular, I'm looking for holomorphic ...
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Can $S^4$ be the cotangent bundle of a manifold?

I am asking the question because in the classical mechanics book by Arnold, he states that there is a distinguished $1$-form on $T^*V$. It seems that there is no such distinguished $1$-form on a ...
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Formal adjoint of curvature (Yang Mills)

Currently reading a paper on finding solutions to the Yang Mills equation $D^*\Omega=0$, where $\Omega$ is the curvature and $D^*$ is the formal adjoint of the exterior covariant derivative $D$. ...
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$V$-bundles and vector bundles

I am looking for more information on $V$-bundles. They are hard to search for as either vector bundles come up or something like GL($V$)-bundles come up. I am looking for some nice expository ...
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What line bundle pulls back to the trivial line bundle

Let $X$ be an abelian surface. $C$ be a curve in $X$. Consider the projective bundle $\pi:\mathbb{P}^1_C\longrightarrow C$. This is a projective morphism. I have two questions : 1) Can we find an ...
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Sheaf of sections of vector bundle over a manifold is an $\mathcal O_M$-module

Section 13.1.2 of Ravi Vakil: "Fix a rank $n$ vector bundle $\pi:V\rightarrow M$. The sheaf of sections $F$ of $V$ is an $\mathcal O_M$-module - given any open set $U$, we can multiply a section over ...
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Vector bundles in Ravi Vakil's notes on quasicoherent sheaves

In chapter 13 (Quasicoherent and coherent sheaves) of Ravi Vakil's wonderful notes, the author starts by discussing vector bundles, supposedly for motivation. Having understood that each locally free ...
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Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - ...
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Möbius band as line bundle over $S^1$, starting from the cocycles

The professor asked us to construct a non-trivial line bundle over $S^1$ by giving an open cover of $S^1$ and the cocycles. My idea was to take as open cover $U_1:=S^1\setminus\{0\}$ and ...
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If a line bundle admits a non-vanishing section then it is trivial

Suppose $\pi:E\to B$ is a line bundle. Let $s:B\to E$ a non-vanishing section, i.e. for every $b\in B$ $s(b)\ne 0$ and $\pi\circ s=Id_B$. I have to prove that the line bundle above is trivial. Idea: ...
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What is the push forward of the canonical class?

Let $X$ be an abelian surface over $\mathbb{C}$. And let $i:X\longrightarrow X$ be the inverse map. $i$ is a degree 2 morphism. We consider $Y$ the quotient of $X$ by the action of $i$, that is, ...
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A fiber product is a fiber bundle

Let $F,B$ be topological spaces. A fiber bundle $E$ over the basis $B$ with fiber $F$ is a topological space $E$ endowed with a continuous surjection $\pi:E\to B$ such that there exists an open cover ...
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Are vector bundles special cases of étale bundles?

Is it possible to define vector bundles as particular instances of étale bundles? An étale bundle is a bundle $p:E\rightarrow X$ which is a local homeomorphism (as in Maclane-Moerdijk): every $e\in ...
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Quick question: Determinant bundle is Cartier?

$X$ is an algebraic surface (i.e. compact complex, which embeds in a projective space). $V$ is a vector bundle of rank 2 over $X$. Why is $\det{V}=\mathcal{O}_X(D)$ for some divisor $D$? Is there a ...
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Does a left group action on a principal bundle induce an action on associated vector bundles?

Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot $ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
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Two definitions of conormal bundle

Suppose $X$ is a smooth manifold and $f: Z \to X$ is an immersion with transverse self-intersection. I've seen (e.g. here) the conormal bundle to $Z$ in $T^* X$ defined as Definition A: $\quad L_Z := ...
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Compactifying $\mathcal{O}_{\mathbb{P}^1} (-2)$

I have the total space of $\mathcal{O}_{\mathbb{P}^1} (-2)$ and I see that a "standard" way to compactify is to add the trivial line bundle, $\mathcal{O}_{\mathbb{P}^1}$, and then projectivize. That ...
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Symbol of the differential operator on vector bundles

Suppose that we have a differential operator $D:C^{\infty}(\mathbb{R}^n) \to C^{\infty}(\mathbb{R}^n)$ of the form $(Df)(x)=\sum_{|\alpha| \leq k}a_{\alpha}(x)\frac{\partial^{|\alpha|}f}{\partial ...
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$w(\mathbb{R}P^q) = 1$ if and only if $q = 2^k - 1$ for some $k$

How do I show that $w(\mathbb{R}P^q) = 1$ if and only if $q = 2^k - 1$ for some $k$?
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Triviality of a vector bundle is an open condition

Is the following statement true (and if not, are there additional assumptions that make it true?) A vector bundle on a variety which is trivial if restricted to a closed subvariety is trivial on ...
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Defining an Ehresmann Connection Using Linear Connection

In some places that I have seen, given the covariant derivative $\nabla$ of a linear connection on a vector bundle $\pi: E\rightarrow M$ (dim(M)=k), an Ehresmann connection is defined in the following ...
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injective morphism between line bundles on curves

Let $X$ be a smooth projective, irreducible, curve, $\mathcal{L}$ be an invertible sheaf on $X$ and $\mathcal{L}' \subset \mathcal{L}$, an invertible subsheaf. Is $\deg(\mathcal{L}') \le ...
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Transitive group actions on Principal bundles

I have a question in regards to page 107 of Kobayashi & Nomizu's Foundations of Differential Geometry Setup: Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$ whose Lie ...
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Transformation behavior of connection on vector bundle.

Using the notation from Jost's various books on geometry, let $$ D=d+A $$ be a connection on a vector bundle $\pi:E\rightarrow M$ with structure group $GL(n,\mathbb{R})$. Also let $\{U_\alpha\}$ be ...
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Determine when $T(S^n \times S^k)$ is a trivial tangent bundle.

My partial answer is that , to make $T(S^n \times S^k)$ trivial, a neccessary condition is that $n$ or $k$ is odd. My argument is as follows: Suppose $T(S^n \times S^k)$ is trivial, that is, $S^n ...
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Classification of rank $\geq 2$vector bundles over Grassmanians

Are there classification results of higher rank (complex) vector bundles over (complex) Grassmanian manifolds? For example, we know that line bundles are in correspondence with the $H^2(G)$, the ...
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If $\xi$ is a vector bundle and $f^{*}(\xi)$ is the bundle induced by $f$ then $f^{*}(\xi)$ is orientable.

Let $\xi = \pi : E \rightarrow X$ be a vector bundle and $f: Y \rightarrow X$ a continuous map. Let $E' \subset Y \times E$ be the set of all $(y,e)$ with $f(y) = \pi(e)$. We can define a vector ...
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Prove that $s: B \rightarrow E$ is a section ( vector bundles)

I'm very very unfamiliar with vector bundles, so maybe this question is quite trivial. Let $\pi : E \rightarrow B$ be a vector bundle and $s: B \rightarrow E$ a map sending each $p \in B$ to the $0$ ...
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Is there a natural Dolbeault operator on a almost holomorphic vector bundle?

For vector bundles $(\pi: V \rightarrow M )$ over a complex manifold, there is a notion of holomorphicity that can be defined in two equivalent ways : $V$ is a complex manifolds and $\pi:V ...
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construction of linear independent local section in vector bundle [duplicate]

Let $X_1,\ldots,X_k$ be lineary independent local sections of a vector bunlde $V$, $rank(V)>k$, defined on some open set $U$. Can I construct a local section $X_{k+1}$ on a (maybe smaller) open ...
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Explaining the definition of vector bundles

Recall the definition of a vector bundle: Let $M$ be a topological space. A $k$-dimensional vector bundle over $M$ is a topological space $E$ with a surjective continuous map $\pi\colon E \to ...
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Why are compact complex manifolds Liouville?

I know this is true but strangely can't find references. Also, consider the trivial $n$-bundle over any connected compact manifold, does Liouville imply that all holomorphic sections are constant? ...
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Proof of the formula for dimension of moduli of stable vector bundles on smooth curves

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...
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$\bigwedge^k T^*M$ is a $\binom{n}{m}$-dimensional Subbundle of $\bigotimes^k T^*M$.

I am trying to prove the following: Let $M$ be a smooth manifold. Then $\bigwedge^k T^*M$ is a smooth subbundle of dimension $\binom{n}{k}$ of $\bigotimes^kT^*M$. To do this, I think the ...
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On a scale of 1 to 10 how far is this manifold from being a normal bundle?

(DIS)-CLAIMER: All the manifolds considered in this post are completely humble and have no additional structure beyond the smooth structure. Let $Y$ be a submanifold of $M$ and let $(-)^0$ denote the ...
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Obstruction to the splitness of an exact sequence of holomorphic vector bundles

This question was asked here, but I think there will not be an answer, so let me recopy the question on Mathoverflow. In the book of S. Kobayashi, hyperbolic complex spaces, there is the lemma ...
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Obstruction to the splitness of an exact sequence of holomorphic vector bundles [duplicate]

In the book of S. Kobayashi, hyperbolic complex spaces, there is the lemma (3.A.3) on the page 153, section Royden's extension lemma, whose statement is Every exact sequence of holomorphic vector ...
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Action of $H^1$ on spin structures

Since the set of spin structures on a principal $SO(n)$-bundle on a manifold $X$ is in one to one correspondence with $H^1(X,Z_2)$, this group admits an action on spin structures. I wanted to know if ...