All Questions
19 questions
4
votes
0
answers
167
views
What textbooks/papers should I read to try to make this rigorous?
Consider a surface of revolution $S$ and an embedding $e:S \hookrightarrow X^3$ for $X^3=[0,1]^3$ with cone points $p,q$ elements of $\partial X^3$ where $\partial X^3=X^3-(0,1)^3$ for $\mathrm {sup}~ ...
- 383
2
votes
0
answers
251
views
The conormal sheaf is the sheaf of sections of the conormal bundle for smooth manifolds
$\def\sO{\mathcal{O}}
\def\d{\mathrm{d}}$In ringed spaces theory, there is a notion of “conormal sheaf of an immersion” (mainly used in scheme theory), whereas in smooth manifold theory, there is the ...
8
votes
3
answers
976
views
Examples and properties of spaces with only trivial vector bundles
Let $B$ be a paracompact space with the property that any (topological) vector bundle $E \to B$ is trivial. What are some non-trivial examples of such spaces, and are there any interesting properties ...
- 1,763
6
votes
2
answers
448
views
About the index theorems
I am looking for some introductory book/paper/notes about the several index theorems and their applications. By several I mean the "classical" Atiyah-Singer theorem, the local index theorem (...
- 789
15
votes
1
answer
672
views
Reference for the Swan-Serre theorem as a monoidal equivalence
Let $X$ be a compact Hausdorff The well-known Swan--Serre theorem gives an equivalence between the continuous vector bundles over a compact Hausdorff space $X$, and finitely-generated projective $C(X)$...
- 393
8
votes
1
answer
823
views
General wedge-product for vector bundle valued forms
In mathematics and physics, especially gauge theory, there are many different but related notions of wedge products when discussing vector space- and vector bundle-valued differential forms. For ...
- 1,429
1
vote
0
answers
310
views
Can we classify two dimensional complex vector bundles over $\mathbb{R}P^2$?
I know it is easy to classify line bundles over $\mathbb{R}P^2$. But do we have a classification of two dimensional complex vector bundles over $\mathbb{R}P^2$?
- 9,019
0
votes
1
answer
108
views
Intersection Grassmanian planes
I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and ...
- 1,043
19
votes
3
answers
2k
views
what is a spinor structure?
There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$,
a Riemannian metric is a section of positive definite symmetric bilinear forms on $TM$
or ...
- 4,432
2
votes
2
answers
2k
views
Commuting of exterior derivative and contraction (vector-valued forms)
$\newcommand{\sig}{\sigma}$
$\newcommand{\tr}{\operatorname{tr}_{\eta}}$
$\newcommand{\al}{\alpha}$
$\newcommand{\be}{\beta}$
$\newcommand{\til}{\tilde}$
Let $E$ be a smooth vector bundle over a ...
- 6,741
6
votes
0
answers
388
views
What’s the limit of a vector bundle?
In geometric measure theory, there’s an answer to the question “what’s the limit of a family of submanifolds”, namely there’s some kind of object called an integral current.
In the geometric ...
- 8,723
4
votes
2
answers
475
views
How to compute the index of such operator?
Let $M$ be a compact Riemannian manifold, with $R$ nowhere-vanishing vector field on $M$(whose orbit may be closed/ not closed). $E$ and $F$ are two vector bundle (Edit: which are sub-bundles of $\...
- 857
4
votes
1
answer
234
views
A trivialization of an almost complex structure
Recently, I have been studying the Carleman Similiarity Principle, which is used to study the regularity and unique continuation of J-holomorphic curves.
Roughly, one takes a solution $ u $ of a ...
- 337
8
votes
2
answers
994
views
Homotopy invariance of vector bundles by parallel transport: reference needed for my students.
Let $M$ be a smooth manifold and $V \to [0,1] \times M$ be a smooth vector bundle. The homotopy invariance states that the restrictions $V_0$ and $V_1$ to the bottom and top of the cylinder are ...
- 20.9k
4
votes
0
answers
367
views
Representation theory and associated bundles
I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
- 6,025
7
votes
1
answer
865
views
Associated vector bundles of infinite rank and induced connections
Let $\mathbb{V}$ be a representation of a Lie group $G$ and let $P \to M$ be a principal $G$-bundle with a principal connection. If $\mathbb{V}$ is finite-dimensional, then one can associate to this ...
- 8,597
2
votes
0
answers
145
views
Semistability of a sheaf on nodal curve
Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...
- 2,323
1
vote
1
answer
252
views
Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles
Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
- 729
7
votes
0
answers
116
views
Bundles over Function Spaces
Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable Hilbert-...
- 71