Tagged Questions
2
votes
0answers
28 views
Curvature form projective spaces
Let $T\mathbb{C}P^n$ tangent bundle over $\mathbb{C}P^n$. We have an hermitian metric on $T
\mathbb{C}P^n$ defined as $h=\frac{dzd\bar{z}}{(1+|z|^2)^2}$. If we consider Levi-Civita connection we can ...
2
votes
1answer
39 views
Is Whitney sum of vector bundle a categorical colimit?
We known that the direct sum of two vector spaces is the categorical colimit of vector spaces. My question is whether Whitney sum of vector bundle is a categorical colimit (in the category of vector ...
1
vote
0answers
18 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 ...
1
vote
1answer
23 views
Positive curvature on holomorphic vector bundles
There must be a mistake in my understanding the definition of positivity for the curvature. Let me summarize:
Let $ (L,\nabla,h) \rightarrow M $ be a hermitian hol line bundle with Chern connection. ...
2
votes
1answer
37 views
Prove that a tensor field is of type (1,2)
Let $J\in\operatorname{end}(TM)=\Gamma(TM\otimes T^*M)$ with $J^2=-\operatorname{id}$ and for $X,Y\in TM$, let
$$N(X,Y):=[JX,JY]-J\big([JX,Y]-[X,JY]\big)-[X,Y].$$
Prove that $N$ is a tensor field of ...
1
vote
1answer
33 views
Distribution and Tangent Bundle
Let $F=\{F_p; p\in M\}\subseteq TM$ be a rank $k$ smooth distribution. Can anyone explain-me what is the set $$\displaystyle\nu(F)=\frac{T_pM}{F_p}.$$
4
votes
1answer
55 views
Vector Bundle Doubt..
Well I have a doubt about a rank $k$ vector bundle. My definition of vector bundle is: A rank $k$ vector bundle is a triple $(\pi, E, M)$ where $E$ and $M$ are smooth manifolds and $\pi:E\rightarrow ...
2
votes
1answer
38 views
Classification of flat complex line bundles
I'm having a contradiction with two different classifications of flat complex line bundles over a manifold $X$. Suppose for simplicity that $H^2(X;\mathbb Z) = 0 = H^1(X;\mathbb C)$. Then the only ...
3
votes
2answers
39 views
Vector Bundles and Distributions
How can I show that following: If $F\subseteq TM$ is a smooth distribution then $F$ is vector bundle and the inclusion $F\hookrightarrow TM$ is a morphism of vector bundles?
3
votes
0answers
30 views
Tangent bundle of a quotient by a proper action
Given a compact group $G$ acting freely on a manifold $X$, is there a "nice" way to describe the tangent bundle of the quotient $X/G$ (when it is a manifold)?
In the case the group $G$ is finite, or ...
2
votes
1answer
60 views
Trivial Tangent and Cotangent Bundles
If we have a smooth manifold $M$, why is the tangent bundle $TM$ trivial (as a vector bundle) iff the cotangent bundle $T^*M$ is trivial as well?
6
votes
1answer
90 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 ...
4
votes
2answers
86 views
Bundle Automorphisms, Structure Groups and Gauge Groups
I am trying to get my head around the mathematical foundations of gauge theory and wanted to check that I am correct in thinking the following is true.
If $E$ is a $G$-principle bundle over $M$ then ...
1
vote
1answer
35 views
Vector space structure on $(-1,1) \subset \mathbb{R}$ (or: möbius strip as vector bundle)
I'm first putting the question into it's context, so probably you can see if i'm asking the wrong question to get what i want.
The Task is to show that the Möbius (Moebius) strip is a Vector bundle ...
8
votes
1answer
91 views
Vector Bundle Over Contractible Manifold
The problem comes from Liviu Nicolaescu's book Lectures on the Geometry of Manifolds. He asks the reader to prove that any vector bundle $E$ over $\mathbb{R}^n$ is trivializable. The idea he gives is ...
3
votes
1answer
99 views
Trivial tangent bundle and orientability
Let $M$ a (real) $n$-dimensional connected differentiable manifold.
(a) The tangent bundle $TM$ is trivial, $TM \simeq M \times \mathbb R^n$;
(b) $M$ is orientable.
Consider the ...
3
votes
0answers
44 views
Questions on a sub-bundle of $\mathbb R^3\setminus \{0\}$
Let us consider spherical coordinates $(r,\theta,\phi)$ on $\mathbb R^3$ and the manifold $M:=\mathbb R^3 \setminus \{0\}$.
Let us consider the 1-form on $M$
$$
\omega = zdz ...
4
votes
2answers
97 views
Describe tangent and normal bundle to a manifold
Consider the set
$X:=\{(x,y,z)\in\mathbb{R}^3 | x^2+3y^2=1+z^2\}$
I have to show that $X$ is a submanifold of $\mathbb{R}^3$ (and this is trivial); then, using on $T\mathbb{R}^3$ the standard ...
1
vote
0answers
47 views
Complex projective manifolds and holomorphic mappings
Let $X\subset\mathbb{P}^2$ be a complex manifold defined by a homogeneous polynomial of degree $d>3$. Let $$\phi:\mathbb{P}^1\rightarrow X$$ be a holomorphic map. Show that $\phi$ is constant.
...
6
votes
0answers
111 views
Isomorphism between spaces of sections.
Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
4
votes
3answers
125 views
What is a tangent bundle? (Aubin)
Here's what I read in A Course in Differential Geometry by Thierry Aubin.
2.5. Definition. The tangent bundle $T(M)$ is $\bigcup_{P\in M} T_P(M).$
And then
2.6. Definition. Let $\Phi$ be a ...
2
votes
0answers
73 views
Complex vector bundles with real transition functions
After doing a bit of playing around (I think) I was able to show that the map $\operatorname{id}\otimes\ \psi : \Omega^{p,q}(X, E) \to \Omega^{p,q}(X, \overline{E})$, where $\psi $ is the conjugation ...
1
vote
1answer
44 views
What is a Fourier decomposition of the index of an operator?
Consider a compact manifold $M$ equipped with some $S^1$-action, and let $E,F$ be vector bundles over $M$. Suppose further that a fixed elliptic operator $D:\Gamma(E)\to\Gamma(F)$ is preserved under ...
1
vote
2answers
168 views
Extending a Set of Linearly Independent Vector Fields to a Basis
My question is this. Suppose we are given some smooth vector fields $X_1, X_2,..., X_k$ which are linearly independent at all points in a neighborhood $U$ (EDIT: diffeomorphic to a ball) of $R^n$. Do ...
0
votes
1answer
88 views
When does the normal vector of a Moebius-strip intersect?
In class the teacher was talking about normal vectors. $r = \langle x,y\rangle$ then the normal vector is
$$ N\left(t\right) = \frac{T^{\prime}\left(t\right)}{||T^{\prime}\left(t\right)||} $$
where ...
8
votes
2answers
135 views
Almost A Vector Bundle
I'm trying to get some intuition for vector bundles. Does anyone have good examples of constructions which are not vector bundles for some nontrivial reason. Ideally I want to test myself by seeing ...
4
votes
1answer
116 views
Existence of Complex Structures on Complex Vector Bundles
Let $E$ be a real vector bundle on a smooth manifold $X$. Let $J : E \to E$ be a vector bundle morphism (i.e. $\pi \circ J = \pi$, where $\pi : E \to X$ is the projection map) with $J^2 = ...
4
votes
0answers
115 views
de Rham Cohomology of Non-Flat Bundle
Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$.
If $E$ ...
6
votes
3answers
221 views
Vector bundle transitions and Čech cohomology
I have read that transition maps $g_{\alpha\beta}:U_\alpha\cap U_\beta\to GL(n)$ of a vector bundle of rank $n$ are related to the Čech cohomology group $H^1\left(M,GL(n,\mathcal{C}^\infty_M)\right)$ ...
4
votes
1answer
84 views
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 ...
5
votes
2answers
115 views
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 ...
1
vote
2answers
104 views
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 ...
5
votes
0answers
174 views
understanding this differential operator on a tensor product
I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
2
votes
1answer
88 views
The Affine Property of Connections on Vector Bundles
Given any two connections $\nabla_1, \nabla_2: \Omega^0 (V) \to \Omega^1 (V)$ on a vector bundle $V \to M$, their difference $\nabla_1 - \nabla_2$ is a $C^\infty (M)$-linear map $\Omega^0 (V) \to ...
6
votes
0answers
148 views
Orientability of the total space of a vector bundle over an oriented manifold
Let $M$ be a (smooth) manifold of dimension $n$, and let $\pi : E \to M$ be a (smooth) vector bundle of rank $r$. If I choose a connection on $E$, then I obtain a decomposition of $T E$ as $V E ...
3
votes
1answer
194 views
The space of smooth sections of a vector bundle.
Let $M$ be a compact, finite-dimensional manifold and $\pi : \mathrm{B} \rightarrow M$ a vector bundle over $M$ whose typical fiber is $\mathbb{R}^n$. Denote by $\mathcal{C}^{\infty}(M, \mathrm{B})$ ...
4
votes
1answer
226 views
The Canonical Bundle over a Riemann Surface
I am trying to understand an example of a line bundle over a Riemann surface; as it is very terse and short, I have lots of trouble. It is written in block-quotes below, and I ask questions as I go.
...
8
votes
2answers
85 views
Explicit formula for the curvaure of a connection
Let $E$ be a vector bundle over $M$ and denote by $\mathcal{A}^k(E)$ the space of sections of $\Lambda^k (TM)^* \otimes E$, i.e. the space $k$-forms with values in $E$.
A connection ...
4
votes
1answer
114 views
Image of smooth vector bundle morphism
Let $\pi_1:V_1 \rightarrow B_1$ and $\pi_2:V_2 \rightarrow B_2$ be smooth vector bundles and write $\phi_V: V_1 \rightarrow V_2$ as well as $\phi_B:B_1 \rightarrow B_2$ respectively for the total and ...
3
votes
1answer
244 views
Tautological vector bundle over $G_1(\mathbb{R^2})$ isomorphic to the Möbius bundle
Let $V$ be a finite dimensional vector space, and let $G_k(V)$ be the
Grassmannian of $k$-dimensional subspaces of $V$. Let $T$ be the
disjoint union of all these $k$-dimensional subspaces and ...
2
votes
2answers
358 views
The pullback $F^\ast :T^*N \rightarrow T^*M$ is a smooth bundle map
How can I show that the pullback $F^*: T^*N \rightarrow T^*M$ associated with $F:M \rightarrow N$ is a smooth bundle map if it is a diffeomorphism?
2
votes
2answers
378 views
Real line bundle smoothly isomorphic to Möbius bundle
I am reading Lee's Introduction to Smooth Manifolds and got stuck on the problem 5.6. The question is written here, question 1. (There is a typo in the question. The last sentence should be "Show that ...
3
votes
2answers
348 views
Understanding differential form
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 ...
3
votes
1answer
199 views
Prove that $TS^{n-1}$ is a trivial bundle, if $\mathbb{R}^n$ may be provided with an $\mathbb{R}$-algebra structure without zero divisors
Here I was shown how to prove that $TS^1$ is a trivial bundle.
Similarly, I can show that $TS^3$ is a trivial bundle.
Identify $\mathbb{H}$ with $\mathbb{R}^4$ and that that for $x \in S^3$ we have ...
13
votes
7answers
536 views
An example of a triple $(E,\pi,M)$ which is not a vector bundle
What is an example of a pair of finite dimensional $C^{\infty}$ manifolds $E$ and $M$, and a smooth function $\pi:E\rightarrow M$ such that $\pi^{-1}(p)$ has a vector space structure for each $p\in M\ ...
0
votes
1answer
250 views
