For questions on vector bundles.

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If the linearization of a section is surjective on a slice, is the image under a submersion also a smooth manifold?

Let $V \rightarrow M \times N_1 $ be a vector bundle, where $M$ and $N_1$ are smooth manifolds and $s: M \times N_1 \rightarrow V$ a smooth section such that if $s(p,q) =0$ then $$ \nabla ...
4
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1answer
27 views

How to prove line bundle L is trivial if and only if its dual bundle us trivial?

How to prove line bundle L is trivial if and only if its dual bundle us trivial ?
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1answer
36 views

Basis for a line bundle

I have a basic confusion. The following is drawn from Huybrechts' Complex Geometry text: "Let $L$ be a holomorphic line bundle on a complex manifold $X$ and suppose that $s_0,\dots, s_n \in H^0(X, ...
2
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3answers
54 views

normal bundle on a submanifold

Can you give me an example of a nontrivial normal bundle of a submanifold (of any manifold)? There is standard example of the core circle of a mobius band, but can you give an example of a submanifold ...
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0answers
8 views

Reference: Differential operators and principal symbols

I am looking for good references about differential-/pseudodifferential operators and principal symbols. thanks
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1answer
30 views

0th-order differential operator vs. 1st-order differential operator on a vector bundle $(E, \pi,M)$

Consider a vector bundle $(E,\pi,M)$. A 0th-order differential operator on $E$ is a $C^\infty(M)$-linear endomorphism $\Gamma E\rightarrow\Gamma E$. $\Gamma E$ is the set of sections on $M$. A ...
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1answer
36 views

Complex structure on a real vector bundle

Let $M$ be a smooth manifold and $\pi:E \rightarrow M$ a real vector bundle and note $E_x:=\pi^{-1}(x), \forall x\in M$. We set a bundle $\text{End}(E)=E\otimes E^*$. Now suppose there exists a smooth ...
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0answers
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Real vector bundles over $S^1$

Prove that for every real vector bundle over $S^1$ there exist open connected subsets $U_1,U_2 \subset S^1$ with $U_1 \cup U_2=S^1$ such that $E$ is trivial over $U_1$ and over $U_2$. I need an ...
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0answers
36 views

Do homotopic transition functions define isomorphic bundles (on smooth manifolds)?

It is fairly known that given a cover $\{U_i\}_{i \in \mathbb{N}}$ of some smooth manifold $M$ together with smooth transition functions $g_{\alpha \beta} \colon U_{\alpha} \cap U_{\beta} \rightarrow ...
3
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2answers
57 views

Action on sheaf cohomology in Bott-Borel-Weil theorem

Let $G$ be simply connected complex semisimple Lie grou and $P \subseteq G$ parabolic subgroup. Suppose $V$ is finite dimensional irreducible representation of $P$ with highest weight $\lambda$, and ...
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0answers
26 views

Computing cohomology of the sheaf $\mathcal{End}(T_{\mathbb{P}^2})$ restricted to a curve

Characteristic of the basic field is zero in this question. Let $E \subset \mathbb{P}^2$ be a smooth elliptic curve. Let $\mathcal{F}$ be the vector bundle on $E$ obtained as restriction to $E$ of the ...
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2answers
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Stably equivalent bundles and stable homotopy classes of maps

We know that there is a one-to-one correspondence between isomorphism classes of principal $G$-bundles over a base space $M$ and homotopy classes of maps $M \to BG$, where $BG$ is the classifying ...
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0answers
23 views

vector bundles and cocycles

I need a detailed solution to a self-study book's exercise: "Show that two vector bundles on M are isomorphic iff their cocycles relative to some open cover are equivalent" I can show it in one ...
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0answers
24 views

Global Trivialization of $M\oplus M$

$\mathbb S^1$'s $\mathbb R^1$-bundle is $$\{\mathbb S^1\times\mathbb R^1\text{, open Mobius strip}\}$$ and its $\mathbb R^2$-bundle is $$\{\mathbb S^1\times\mathbb R^2\text{, open solid Klein ...
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1answer
19 views

“Restriction” of a Differential Operator on a Vector Bundle to the Space of Local Sections?

I'm trying to understand the definition of differential operators on vector bundles. The material I'm following http://www.mat.univie.ac.at/~stein/research/talks/nhops.pdf starts with the ...
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0answers
13 views

Reference request: Order of a vector bundle

Please could you link me to an accessible reference/set of lecture notes on the definition of the order of an algebraic vector bundle? Google just shows some very complicated definition which is ...
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0answers
28 views

homotopy via isotopy

Is it possible to realize a homotopy between two vector bundles (which are sub-bundles of the tangent bundle) over the same base $B$ as induced by an ambient isotpy? In other words, assume $X_t $ is a ...
2
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0answers
52 views

Why is this map of sheaves surjective?

Let $M$ be a complex manifold. Let $\mathcal{L}$ be a holomorphic line bundle over $M$, with $\pi:M\longrightarrow\mathcal{L}$ being the projection map. Any section $s$ of $\mathcal{L}$ over $M$ is a ...
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1answer
24 views

Defintion of Generalized Conic Bundle

Can some one help me understanding why the Definition of Generalized Conic Bundle is generalization of the Conic Bundle definition. This is the definition of a conic bundle from "Comparison theorems ...
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0answers
16 views

trivialization of a bundle

suppose we have a $D^2$-bundle $X$ over a surface with boundary $F$. It is said that we can always trivialize the bundle: X is diffeomorphic to $F\times D^2$ but I do not see why this is true.
2
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1answer
23 views

Holonomy representation: is it actually a class of representations?

In D. Joyce's book "Riemannian Holonomy Groups and Calibrated Geometry" (2007) the author writes that if $\nabla$ is a connection on a vector bundle $E$ (over a connected base) with the fibre $\mathbb ...
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1answer
13 views

Multiple points in the parallel transport equation

Let $E \to M$ be a vector bundle with connection $\nabla$, $\gamma \colon [0,1] \to M$ a smooth curve. A section $s = \gamma^* \tilde s$, $\tilde s \in \Gamma(E)$ of $\gamma^* E$ is called parallel if ...
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0answers
36 views

On the Chern connection

It is well kown that If $\,\bar\partial_E$ defines a holomorphic structure on the complex vector bundle $E \to X$ and $K$ is a hermitian metric on (the fibres of) $E$, there is a unique connection ...
4
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2answers
59 views

Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ ...
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0answers
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Elements in the zero subbundle

I am in the following situation: There is a vector bundle $W$ with basis $\Omega$ and a flow on $W$, i.e. a function $\Psi\colon W\times\mathbb{R}\to W, (w,r)\mapsto w\cdot r$ that fullfills the ...
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0answers
41 views

Tangent Bundles to manifolds

I am having trouble trying to visualize exactly what a tangent bundle to the klein bottle is spuposed to look like. Is it possible for one to decompose it as a direct sum of simpler bundles?
2
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1answer
36 views

Homotopy groups of infinite Grassmannians

Let $G_k$ be the infinite rank $k$ Grassmannian. For $n>k$, is $\pi_n(G_k)$ trivial? Phrased differently, is every rank $k$ vector bundle on an $n$-sphere, for $n>k$, trivial? (This is motivated ...
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0answers
37 views

Finding global sections of a vector bundle over a compact manifold which generate each fiber

The following is an exercise I am doing for review for a midterm exam in differential geometry. Question: Let $\pi:V\to X$ be any real differentiable vector bundle of rank $r$. Assume that $X$ is ...
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0answers
8 views

Vector bundle base space map

Is it true, and if it is, is there some easy way to see the following? Suppose that $\xi = (\pi, E, B)$ is an $n$-vector bundle with $B$ paracompact but not necessarily compact. Is there a base ...
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0answers
13 views

Embedding into the projective space - tangent level injection

I am reading Griffiths and Harris, the section of embedding a manifold $M$ into projective space. Let $\mathcal{L}$ be a line bundle over $M$, with dim $H^0(M,\mathcal{L})=N$, and let $s_0,...,s_N$ be ...
3
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1answer
50 views

Canonical isomorphism between vector bundle and dual?

So, we've been asked to show, given a real vector bundle equipped with a metric, that there is a canonical isomorphism from the vector bundle and its dual. Now, there's a theorem that says two vector ...
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1answer
39 views

Let $\pi: E \to M$ a vector bundle. Is $E$ a direct summnad of $M\times\mathbb{R}^{d}$, for some $d$?

Let $\pi: E \to M$ a vector bundle over a smooth manifold $M$. $E$ is direct summand of $M\times\mathbb{R}^{d}$, if there exist a vector bundle morphisms $f:E\to M\times\mathbb{R}^{d}$ and ...
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1answer
25 views

Smooth structure on a quotient vector space

How do I know if $$f:\mathbb{R}^n/(\mathbb{R}^k\times \{\mathbf{0}^{n-k}\})\to \mathbb{R}^n/(\mathbb{R}^k\times \{\mathbf{0}^{n-k}\})$$ is smooth? Can't find the definition of the canonical smooth ...
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1answer
42 views

A Question on Vector Bundles

I am having trouble proving the following about vector bundles. I would think it would be rather easy, but I can't think of how. This isn't homework, but something I want to be true so that I can ...
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1answer
52 views

Quick question: a 2:1 map onto the projective line

Given a line $L$ in $\mathbb{P^2}$. How do we see that a surjective map $\mathcal{O}_\mathbb{P^2}^{\oplus2}\rightarrow j_{*}{\mathcal{O}_L(2)}$ ($j$ is the inclusion of $L$ to $\mathbb{P^2}$) ...
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0answers
49 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
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0answers
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Characterizing the Slice Chart of a Subbundle

A rank-$k$ subbundle $F$ of a rank-$n$ smooth vector bundle $E$ is a vector bundle which is smoothly embedded in $E$, whose intersection with a given fiber of $E$ is a subspace of that fiber. Can we ...
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1answer
83 views

Is a space with no nontrival vector bundles contractible?

Let $X$ be a "nice" space, say having the homotopy type of a CW complex. Suppose also that $X$ is connected. Suppose that all real vector bundles on $X$ are trivial. Does it follow that $X$ is ...
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0answers
22 views

$d+\Gamma$-understanding connections

$d+\Gamma$ is a formula commonly found in textbooks when one reads about connection. To be more precise: if $U$ is an open set over which a given vector bundle $E \to M$ is trivial then a coonection ...
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0answers
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How many charts?

My question is rather vaque but I hope that You will feel what I would like to know. A manifold (say: real, $n$-dimensional) is something which locally looks like an open set in $\mathbb{R}^n$. A rank ...
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0answers
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Determining an explicit line bundle over surface

The following is a explicitly defined complex line bundle $E\to\Sigma$ over a closed surface: View $\Sigma$ as a subset of $\mathbb{R}^3$ and consider its Gauss map $n:\Sigma\to S^2$ given by ...
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0answers
33 views

Condition for the generic vector bundle to be globally generated

In the paper "Gwena, Teixidor - Maps between moduli spaces of vector bundles and the base locus of the theta divisor", it is stated without proof that a $general$ vector bundle on a curve of rank $r$ ...
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Frobenius theorem for singular foliations , what are hypotheses impose over an endomorphism $P:TM\to TM$ that it span a singular foliation in M?

Let $M$ a smooth manifold. Given a morphism $P:TM\to TM$, i.e, $\pi\circ P\equiv Id$ and is linear over the fibers. If we suppose that $P^2=P$, then for each $x\in M$, $P_{x}:T_xM\to T_xM$ is a ...
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0answers
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The second cohomology of total space of the $\mathbb CP^1$ bundle

$X$ is a closed smooth surface with $L$ a complex line bundle on $X$. Consider the $\mathbb CP^1$-bundle $P(L\oplus 1)$, that is the projectivization of the sum of $L$ and the trivial line bundle on ...
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0answers
36 views

Holomorphic line bundle over complex torus.

Let $X$ be a complex torus, given by $X = \mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$. How to specify a holomorphic line bundle over $X$? One standard way is to glue it together from trivial bundles ...
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1answer
51 views

computing transition function of tangent bundle $S^n$

I'm just starting to learn about vector bundles, I want to compute the transition functions of the bundle $TS^n$. I started with the stereographic atlas $U_1 = S^n - \{N\}$ and $U_2 = S^n - \{S\}$ ...
2
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1answer
37 views

Trivial line bundle

Suppose that $L \to M$ is complex line bundle over a manifold $M$. One can therefore form the dual bundle $L^* \to M$. We can identify $L^* \otimes L$ with endomorphism bundle $End(L)$. Why it is true ...
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1answer
60 views

Non-vanishing differential forms

Let $M$ be a differentiable manifold of dimension $n$. If the tangent bundle is trivial, then the cotangent bundle is trivial, and so are its exterior powers. In other words, on a parallelizable ...
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1answer
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Complex bundles on $S^{2n+1}$

We know that the complex $K$ theory of spheres are 2 periodic. On the other hand every complex bundle on $S^{1}$ is trivial. So $K(S^{2n+1})=0$. So this is a motivation to ask: Is there a ...
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1answer
52 views

Real vector bundles on $S^{7}$

Is it true that $\pi_{6}(O(n))=0$ for all n? Equivalently, are all real bundles on $S^{7}$ trivial?