For questions on vector bundles.
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 $E_1$, ...
6
votes
0answers
151 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 ...
5
votes
0answers
93 views
Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both mobius strips?
This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!)
I have been working on giving ...
5
votes
0answers
75 views
When is the pushforward / direct image of a reflexive sheaf locally free?
I have seen a number of theorems that guarantee the direct image of a reflexive sheaf to be reflexive again, or for the direct image of a locally-free sheaf to be locally free again.
This makes me ...
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 ...
4
votes
0answers
45 views
How should we think of 'differences' of vector bundles?
Given a Hausdorff space $X$, the set of all vector bundles of finite dimensions $Vect(X)$ (up to isomorphism) with respect to the direct sum is a monoid. We can use the group completion construction ...
4
votes
0answers
39 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 ...
4
votes
0answers
116 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 $E$ ...
4
votes
0answers
90 views
Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?
Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that ...
4
votes
0answers
70 views
Decomposition of vector bundles over a CW complex
Let $X$ be CW complex having only cells up to dimension $n$.
I want to prove that every $m$-dimensional vector bundle $E$ over $X$ decomposes as a sum $E\cong A\oplus B$ where $A$ is a ...
3
votes
0answers
24 views
Maps between total spaces of holomorphic vector bundles
I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles.
Let me outline a situation that is a bit more concrete, to help focus ...
3
votes
0answers
62 views
How to calculate characteristic classes of tensor products?
I was given the following as an execrise:
Prove that the tensor product of complex line bundles over $X$ satisfies the following relationship:
$$c_{1}(\otimes E_{i})=\sum c_{1}(E_{i})$$
It is ...
3
votes
0answers
33 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 ...
3
votes
0answers
42 views
Two questions on jet bundles
I am working with jet bundles on compact Riemann surfaces. So if we have a line bundle $L$ on a compact Riemann surface $C$ we can associate to it the $r$-th jet bundle $J^rL$ on $C$, which is a ...
3
votes
0answers
39 views
Triviality of the tangent space of an abelian variety
The tangent bundle of an abelian variety $A/K$ is trivial. Mumford's proof of this fact goes something like this: given the (non-zero) value of a section in a fiber, one can determine the value in all ...
3
votes
0answers
75 views
A question of extension of vector bundles.
Fix $p \in \mathbb{P}^1$. Let $X=\mathbb{P}^1\times \mathbb{P}^1$, $C_1=\mathbb{P}^1\times \{p\}$ and $C_2=\{p\}\times \mathbb{P}^1$. Since $\mathrm{Ext}^1(\mathcal{O}_{C_2},\mathcal{O}_{C_1})\cong ...
3
votes
0answers
39 views
Any vector bundle on $\mathbb R$ is a trivial bundle
How to prove that any vector bundle on a Euclidean space is a trivial bundle? It is enough to prove it for the case of dimension $1$ and I hope it will be a nice exercise for me to generalize to 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 $M$
$$
\omega = zdz ...
3
votes
0answers
78 views
Alternate pullback bundle construction
If $\pi : F \to N$ is a fiber bundle, and $\phi : M \to N$ (here, $M$ and $N$ are manifolds), then the standard way to define the pullback bundle of $F$ by $\phi$ is
$$\phi^* F := \{(m,f) \in M ...
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
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 ...
2
votes
0answers
50 views
Second order equations on manifolds
In my notes, the lecturer considers a smooth vector field $v: TM\to T(TM)$, with $M$ a smooth manifold. Let's write
$$v(u,e)=((u,e), (a(u,e),b(u,e)).$$
It is said that $v$ is a second order equation ...
2
votes
0answers
42 views
extending trivializations of plane bundles via obstruction theory
I came across the following statement: A trivialized $k$-plane subbundle of a trivialized $(n+k)$-plane bundle determines a trivialization of the orthogonal $n$-plane bundle over the $(n-1)$-skeleton ...
2
votes
0answers
65 views
Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?
Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
2
votes
0answers
37 views
How to understand the coordinate transition map in $E\otimes E'$?
This question seems embarrassingly basic. Let $X$ be some manifold and $E,E'$ two vector bundles over $X$ with rank $n$ and $n'$ respectively. Now assume to possible refinement we have $E,E'$ defined ...
2
votes
0answers
64 views
Example of excess intersection theory?
Let $M$ be a smooth manifold of dimension $m$ and $\pi:E\rightarrow M$ a vector bundle of rank $e$. Given a section $s$ of the bundle $\pi:E\rightarrow M$, we expect that the zero locus $Z(s)$ of $s$ ...
2
votes
0answers
58 views
Is Transversality invariant by losing Dimension?
Setup Let $X$ be a smooth, reduced and irreducible scheme. And let $E$ be a vector bundle of rank $n$ on $X$. Following Eisenbud and Harris we say that the collection $\{\sigma_1,...,\sigma_k\}$ of ...
2
votes
0answers
45 views
Possible restriction on first nonzero Stiefel-Whitney classes?
I've been reading Hatcher's book on vector bundles and I'm just getting into the section of Steifel-Whitney numbers. Naturally, I'm interested in which sequences of such numbers are realizable, and ...
1
vote
0answers
24 views
Non-(stable)-triviality of the tautological bundles
The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann manifold $G_k(\mathbb{K}^N)$ of all $k$-planes in $\mathbb{K}^N$ (for $\mathbb{K} = \mathbb{R}$, $\mathbb{C}$ or ...
1
vote
0answers
31 views
Representation of Homogeneous vectorbundle = Induced representation
Hello friends of mathematics :)
I have a question about the induced representation. Suppose $G$ is a group and $H$ a subgroup of $G$. Suppose $\rho$ is a representation of $H$ on the vectorspace $V$, ...
1
vote
0answers
41 views
Canonical bundle and Möbius bundle
I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Möbius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
1
vote
0answers
19 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
0answers
54 views
A funny condition for ampleness on a curve
Let $E$ be a locally free sheaf on a complete nonsingular curve $C$ over $\mathbb C$.
Suppose that, for all points $P$ in $C$, the locally free sheaf $$ E\otimes \mathcal{O}_C(P)$$ is an ample ...
1
vote
0answers
33 views
Nontriviality of pullback of tangent bundle on the Möbius bundle
Consider the Möbius strip $M$ as a vector bundle over $\mathbb R$, in which it is embedded as the zero section. We have the tangent bundle $TM$ over $M$. How to prove that the pullback of this to ...
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.
...
1
vote
0answers
56 views
Complexification of a complex bundle.
Suppose that $E$ is a complex bundle already. This is to say that is can be viewed as a real bundle with an almost complex structure $j$. Then Taubes assert $E$ sits inside its complexification ...
1
vote
0answers
78 views
The Harder-Narasimhan filtration with inverse slopes.
Let $C$ be a complex curve. Recall that the slope of a coherent sheaf $\mathcal{E}$ is defined by
$$
\mu(\mathcal{E})=\mathrm{Arg}(-\mathrm{deg}(\mathcal{E})+i\mathrm{rank}(\mathcal{E}))\in(0,\pi].
$$
...
1
vote
0answers
175 views
Characterization of Chern classes and Whitney product formula
Let E be a Complex vector bundle of rank r on M. Given $s_1, \cdots , s_r$ generic global sections, i can characterize the i-th Chern class as follows:
$ C_i(E)= \eta_{V_i}$, and $V_i$ is the locus ...
1
vote
0answers
264 views
Trivial tangent bundle and parallelizability of a $n$-sphere
So, I want to show that a $n$-sphere $S$ is parallelizable iff it has trivial tangent bundle.
For "$\Leftarrow$" I would like to take a trivialization $\varphi:S\times \mathbb{R}^n\rightarrow TS$ and ...
0
votes
0answers
17 views
Isomorphism canonical and Moebius bundle.
I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Moebius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
0
votes
0answers
30 views
Isomorphy of vector bundles over a variety $X$
As in my other question, let $X$ be a variety over a field $k$, and let $\pi:F\to X$, $\psi:G\to X$ be vector bundles of rank $r$ over $X$ defined on the same open cover $\{U_i\}$. That is, we have ...
0
votes
0answers
35 views
A good way to embed a manifold in a Euclidian space $\mathbb{R}^n$
We know that any closed manifold $X$ can be embedded into some Euclidian space $\mathbb{R}^n$ for sufficiently large $n\in \mathbb{N}$. What is the easiest way to see this fact? I have seen several ...
0
votes
0answers
50 views
Why a section of $\otimes_{k}E^{*}$ defines a $k$-linear, fiber preserving map from $\oplus_{k}E$ to $M\times \mathbb{R}$?
I am not sure if this is a duplicate. Clifford Taubes assert in his book Differential Geometry that we may view sections of vector bundles as homomorphisms from $M\times \mathbb{R}$ to $E$ such that ...
