For questions on vector bundles.

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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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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?
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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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Deforming unstable vector bundle to stable one

Let $V$ be a holomorphic vector bundle. If $V$ is strictly semistable, can we deform $V$ to a stable vector bundle, where stability is defined via the slope function?
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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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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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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 ...
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1answer
43 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
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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
21 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
40 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
51 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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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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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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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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$d+\Gamma$-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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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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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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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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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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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
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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 $U_2 = S^n - \{S\}$ ...
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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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Non-vanishing differential forms and flatness

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
21 views

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
49 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?
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Universal property of tensor product of vector bundles

To define the tensor product of vector bundles $\xi_1$ and $\xi_2$ over base $B$, Milnor-Stasheff's Characteristic Classes takes the space $\sqcup_{b \in B} F_b(\xi_1) \otimes F_b(\xi_2)$ and ...
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1answer
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spin structure definition

Suppose we have a principal $SO(n)$-bundle $E$ over $B$, with projection map $p$. We say that it admits a spin structure if there is a prinicipal $spin(n)$-bundle $E'$ over B, with projection map ...
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Sections of endomorphisms of a vector bundle

The following could be rather silly question but I haven'd found it stated explicitly; from the other side, it seems to me, that this fact is used often without comments. The problem is the ...
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1answer
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Tangent bundle of a tubular neighborhood

Let $N \to X$ be normal bundle of a submanifold $X$ of $Y$. How can I prove that $TN|_{TX}$ is isomorphic to the normal bundle of the inclusion $TX\to TY$? And why this vector bundle is isomorphic to ...
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1answer
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Can we measure how close a vector bundle is to being trivial?

For a vector bundle $E$, I will denote the maximum number of linearly independent global sections of $E$ by $\eta(E)$. We have $\eta(E) \in \{0, 1, \dots, \operatorname{rank}(E)\}$ and $\eta(E) = ...
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1answer
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Is T($S^2 \times S^1$) trivial?

How would I find out if T($S^2 \times S^1$) is trivial or not? Using the hairy ball theorem I can show that T($S^2$) is not trivial, and it is straight forward to show that T($S^1$) is trivial. ...
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2answers
105 views

How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...
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2answers
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Definition of a coordinate vector bundle

Consider the following definition of a coordinate vector bundle. Let $M$ be a smooth manifold of dimension $m$, and $\{(f, U_f)\}$ an atlas of compatible charts for $M$. A smooth coordinate ...
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Invariant characterization of vector bundles associated to a principal bundle?

I have two related questions. Suppose I have a principal $G$-bundle $P\xrightarrow{\pi} M$. The usual construction of an associated vector bundle goes as follows. Fix some representation $\rho : ...
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Do the induced metrics on the dual/tensor product bundle behave well with each other?

Let $E,F$ be complex (holomorphic) vector bundles over a smooth complex manifold $M$. Assume $E$ and $F$ are equipped with Hermitian metrics $h$ and $k$. This induces a metric on $E\otimes F$ namley ...
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1answer
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Sheaf of sections vanishing at a point is $\Gamma(E) \otimes I_p$

Notation: $E$ is a vector bundle with sheaf of sections $\Gamma(E)$. $I_p$ is the sheaf of regular functions on the base space vanishing at a point $p$. $\Gamma_p(E)$ is the sheaf of sections of $E$ ...
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Definition of the $\Bbb C^*$-weight of a line bundle

I'm new to geometric invariant theory and am unsure about a definition. Let $X$ be a smooth projective-over-affine variety equipped with a $\Bbb C^*$ action and let $Z$ be the fixed locus of this ...
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1answer
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Quotient of locally free sheaf is locally free?

If $0\rightarrow F\rightarrow G\rightarrow H \rightarrow 0$ is an extension of $\mathcal{O}$-modules with $F$ and $G$ locally free (each of constant finite rank, i.e. vector bundles), then is $H$ ...
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1answer
38 views

K-theory product trivial on $K^1$ due to stability?

Let $K^0$ be the topological complex vector bundle $K$-theory functor. On the one hand, we have that the product on $K^0(\Sigma X)$ vanishes by a Mayer-Vietoris argument. On the other hand, we define ...
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Vector bundle, nonexistence of Euclidean metric

Milnor-Stasheff "Characteristic classes" problem 2-C says: Any vector bundle over a paracompact base space can be given a Euclidean metric in other words, if $\pi : E \rightarrow B$ is a vector ...
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1answer
111 views

Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
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1answer
106 views

Characteristic Class with arbitrary coefficient

I'm looking for some "natural definition" for characteristic class with arbitrary coefficient. For example, the chern class satisfies four conditions and Stiefel-Whitney class satisfies three ...
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Connections on principal bundles and vector bundles

In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated: 1) Via horizontal subspaces, ...
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Are (certain) metric-preserving vector bundle maps proper?

Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map). Can we ...
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1answer
60 views

Group of Automorphism of the Fiber of Fiber Bundle

Let us consider the Mobius Bundle. Can Someone Explain the part "There is no natural unique homeomorphism of $Y_x$with $Y$. However there are two such which differ by the map $g$ of $Y$ on itself ...
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1answer
63 views

Lifting vector bundles along thickenings

Let $X_0 \to X$ be a nilpotent closed immersion of schemes. Is every vector bundle on $X_0$ the pullback of a vector bundle on $X$? The answer is yes when $X$ is affine. In general, there may be ...
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Well-definedness of the action of the structure group of a principal bundle on the total space.

Find the definition of a fiber bundle here- Definition of Fiber Bundle I am having difficulty in proving that the natural action of $K$ on $X$ is well-defined: Let us recall how does K acts on X ...
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Principal Bundle- Definition cum Exercise from “Geometry and Topology” by Bredon

The definition of fiber bundle can be found from here: Definition of Fiber Bundle Then Bredon defines Principal bundle in the exercise as follows: I am not able to show how K acts naturally on ...