All Questions
19 questions
13
votes
1
answer
1k
views
Atiyah-Singer for pseudodifferential operators via heat kernel?
The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. ...
- 2,998
10
votes
2
answers
2k
views
Atiyah Singer index theorem and Hodge de Rham operator
When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ (...
- 9,330
10
votes
0
answers
409
views
Between Being a Connection and Being an Elliptic Operator
Let $E$ be a smooth complex vector bundle over a smooth compact manifold $M$ and let $H$ be the $Z_{2}$-graded (with the $Z_{2}$-grading given by even/odd forms) Hilbert space of $L_{2}$ (with respect ...
- 505
8
votes
0
answers
112
views
Euler number of the complex of basic forms
Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.
I am trying to understand the Hopf bundle ($G=...
- 181
7
votes
0
answers
253
views
Applications of the Atiyah-Patodi-Singer eta-function $\eta(s)$
The eta function of a differential operator was used by Atiyah, Patodi and Singer to derive their famous index theorem, and is given by
$$
\eta(s)=\sum_{\lambda\neq 0}(\mathrm{sign}\lambda)|\lambda|^...
- 91
6
votes
1
answer
351
views
Fredholm theory of non elliptic operators
In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in ...
6
votes
1
answer
629
views
Index of a differential operator between trivial bundles.
Let $M$ be a closed parallelizable manifold and $D: \Gamma(E) \to \Gamma(F)$ an elliptic differential operator between trivial vector bundles $E,F \to M$. The Atiyah Singer index theorem implies that ...
- 3,244
5
votes
0
answers
280
views
Was an index theorem for manifold with local boundary condition proven?
I would like to ask a question on the bibliography of the index theorems on manifold with boundary.
Before my bibliographical research my understanding of the field was that for manifold with boundary,...
- 51
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
261
views
Lifting a differential operator
Let $D$ be a differential operator acting between the spaces of smooth sections of two vector bundles $E,F$ over compact manifold $M$. If $M$ is not simply connected one can construct the universal ...
- 9,330
3
votes
1
answer
118
views
Index formula for elliptic operators acting on Sobolev sections vanishing on the boundary (say $D: H_0^k(\Omega) \to H_0^{k-1}(\Omega)$)
Given a first order elliptic operator $D:\Gamma(X; E)\to \Gamma(X; F)$ where $X$ is a closed manifold, and $E\to X, F\to X$ vector bundles, we know that $D$ induces a Fredholm operator
between the ...
- 2,533
3
votes
1
answer
344
views
derivative of the adiabatic limit of the eta invariant
To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if ...
- 1,173
3
votes
0
answers
86
views
Some questions on defining the analytic index
The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the ...
- 1,173
2
votes
1
answer
144
views
Perturbation of the adiabatic limit
Let $(M,g_M)$ be a closed oriented Riemannian manifold that has a fibration structure
$$
Y \rightarrow M \overset{\pi}{\rightarrow} B
$$
where $(Y,g_Y)$ and $(B,g_B)$ are closed Riemannian manifolds ...
- 2,535
2
votes
0
answers
91
views
A quantity associated to a compact Riemannian manifold with boundary(The pair of Laplacian)
Let $M$ be a Riemannian manifold with boundary $\partial M$.
Are the following operators, Fredholm operators?Is there a geometric terminology and geometric interpretation for the fredholm index of ...
- 356
1
vote
1
answer
157
views
The index of certain differential operator on tori
Assume that $J$ is an almost complex structure on torus $\mathbb{T}^2$. Let $X$ be a non vanishing vector field on the torus. Let $g$ be a Riemannian metric with corresponding $LC$ connection $...
- 356
1
vote
0
answers
59
views
Ellipticity of certain differential operator associated to a pair of vector field via curvature tensor
What is a precise example of the following situation:
A compact Riemanian manifold $M$ admits two vector field $X,Y$ such that the the operator $$Z\mapsto R(X,Y)Z$$
Would be an elliptic operator and ...
- 356
1
vote
0
answers
308
views
A differential operator associated with a vector field on the torus
Assume that $X$ is a non vanishing vector field on the torus $\mathbb{T}^2$.
We define two linear operators $T,S$ on the space of smooth functions on $\mathbb{T}^2$ as follows:
$T(f)=...
- 356
0
votes
0
answers
62
views
A uniform upper bound for Fredholm index of Laplace quasi-operators on a compact parallelizable manifold
Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...
- 356