For questions on vector bundles.

learn more… | top users | synonyms

2
votes
1answer
33 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
votes
0answers
25 views

Compute with the Tangent Bundle to a projective bundle over projective plane

Suppose I have a vector bundle $E$ on $\mathbb P^2$ and form the projective space bundle $X:=\mathbb P(E)$. Hartshorne (p253) gives an exact sequence to compute $\Omega_{X/\mathbb P^1}$, but I am ...
10
votes
2answers
178 views

Ways to think about vector bundle

I'm studying manifold theory and I've got to the point of discussing the definition of a vector bundle. The definition is quite long and a bit confusing and I was wondering if someone with a bit more ...
2
votes
1answer
30 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 ...
1
vote
1answer
82 views

Complex projective manifolds and holomorphic mappings

Let $X\subset\mathbb{P}^2$ be a complex manifold defined by a homogeneous polynomial of degree $d>3$. Let $$\phi:\mathbb{P}^1\rightarrow X$$ be a holomorphic map. Show that $\phi$ is constant. ...
4
votes
0answers
34 views

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 ...
1
vote
1answer
47 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?
0
votes
1answer
19 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 ...
2
votes
0answers
35 views

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 ...
0
votes
0answers
22 views

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 ...
1
vote
1answer
28 views

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 ...
2
votes
1answer
95 views

Bott connection

Can anyone help me showing the following: Let $E$ be a smooth vector bundle over $M$ and $F\subseteq TM$ a smooth distribution. A $F$-connection is a $\mathbb R$-bilinear aplication ...
0
votes
0answers
27 views

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 ...
0
votes
1answer
40 views

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 ...
3
votes
1answer
67 views

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) = ...
6
votes
1answer
98 views

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. ...
4
votes
2answers
100 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 ...
1
vote
2answers
31 views

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 ...
5
votes
1answer
164 views

Which is the correct universal line bundle: the tautological bundle or its dual?

With topological line bundles over $\mathbb{C}$, one learns that every line bundle is a pullback of the universal line bundle, which is the tautological line bundle over $\mathbb{C}P^\infty.$ In ...
2
votes
1answer
35 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 ...
1
vote
0answers
22 views

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 : ...
3
votes
0answers
46 views

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 ...
2
votes
1answer
37 views

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 $E$ ...
0
votes
1answer
63 views

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 $H$ ...
0
votes
0answers
13 views

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 ...
1
vote
1answer
108 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 ...
3
votes
1answer
86 views

Learning Fibre Bundle from “Topology and Geometry” by Bredon

Bredon defines bundle projection in the following way: $\bf13.1.$ Definition. Let $X,B$ and $F$ be Hausdorff spaces and $p:X\to B$ a map. Then $p$ is called a bundle projection with fiber $F$, if ...
9
votes
5answers
528 views

Reference request: Chern classes in algebraic geometry

I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean. I am looking for a reference that ...
3
votes
1answer
104 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 ...
3
votes
0answers
49 views

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, ...
0
votes
0answers
19 views

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 ...
1
vote
1answer
56 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 ...
4
votes
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 ...
1
vote
0answers
39 views

About the geometric interpretation of the reduced external product in K-theory

I'm trying to fill the details of the explanation of the interpretation of the reduced external product. I'll follow Hatcher second books, and I'll write part of his explanation in the gray boxes ...
1
vote
1answer
28 views

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 ...
1
vote
1answer
93 views

R-linear functionals on manifolds

Surely the following is well known: Let $X$ be a (differentiable) manifold, $R$ the ring of continuous/smooth real functions on $X$, $V$ the $R$-module of all continuous/smooth vector fields on ...
1
vote
1answer
52 views

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 ...
3
votes
1answer
46 views

The long exact sequence in reduced K-theory. How to glue together the short ones?

I'm trying to filling the details about the construction of the long exact sequence in K-theory. I'm using the notation of Hatcher's book (pages 52-53). here is a related question, even thought it's ...
4
votes
1answer
262 views

Global sections of a tensor product of vector bundles on a smooth manifold

This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with ...
6
votes
1answer
145 views

Tensor product of real line bundles is trivial as a map $\mathbb{R}P^\infty\to\mathbb{R}P^\infty\times\mathbb{R}P^\infty\to\mathbb{R}P^\infty$

The tensor product of a real line bundle with itself is trivial, as is easily seen by looking at the transition functions or checking the Stiefel-Whitney class. Real line bundles are classified by the ...
8
votes
2answers
226 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
1
vote
0answers
36 views

Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$ \phi: M \to N. $$ As far as I understand, this gives rise to two distinct isomorphisms $$ a : \mathcal A^1(M) \cong \mathcal A^1(N) :b $$ between the space of ...
2
votes
1answer
53 views

“Visual” interpretation of the Bott Periodicity for complex vector bundles

I've just read the Bott Periodicity proof (complex case) in Hatcher book about K-theory. At first it seems that using this theorem I could conclude that every complex vector bundles over $S^{2n+1}$ ...
0
votes
1answer
17 views

Linear clutching function properties

I'm trying to prove proposition $4.8$ of Husemöller (pay 148) Here it is the text: Where the only hints are: $[L^{n+1}(\xi),L^{n+1}(zp)] \approx [L^{n}(\xi),L^{n}(p)] \oplus [\xi,z]$ ...
0
votes
3answers
51 views

The difference between a fiber and a section of a vector bundle

If $ E_x := \pi^{-1}(x) $ is the fiber over $x$ where $(E,\pi,M)$ is the vector bundle. And the section is $s: M \to E $ with $\pi \circ s = id_M $. This implies that $\pi^{-1} = s $ on $M$. So then ...
2
votes
0answers
35 views

How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
2
votes
0answers
20 views

Euler class of quotient bundle of real projective space

Let $\gamma$ be the real tautological line bundle over the real projective space $\mathbb{R}P^n$, $V$ be the trivial bundle of rank $n+1$, and $Q$ be the quotient bundle $V/\gamma$. What is the Euler ...
4
votes
1answer
69 views

Why doesn't a metric give an isomorphism $TX \cong T^*X$?

Any smooth manifold $X$ admits a Riemannian metric $g$, and we have a map $$ TX \to T^*X, \qquad (x, v) \mapsto (x, g(v,-)) $$ which is smooth if $g$ is. Why isn't this an isomorphism of vector ...
2
votes
1answer
39 views

Working out the details of example 1.13 Hatcher: $\ E_{fg}\oplus n \approx E_f \oplus E_g$

I'm studying an example provided by Hatcher in his K-theory and Vector Bundle book. I'm referring to example 1.13 pag. 24 The first part is clear, $z^2$ is the Kronecker Product of the two ...
3
votes
1answer
65 views

Are vector bundles just modules over $C^{\infty}(M)$, or are “locality” conditions required?

In this question the asker defines 1-forms on a (real, smooth) manifold $M$ to be $C^{\infty}(M)$-module homomorphism[s] from $Vec(M)$ to $C^{\infty}(M).\:\:\:\:(*)$ I'm wondering if this is ...