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7
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0answers
273 views

Directed arcs on a surface

This question is a little odd. I have specific class of structures on a surface, which satisfy several nice properties, and I want to know if they are more natural geometric structures in disguise ...
5
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0answers
164 views

Relative index theorem for Clifford linear Dirac operators

Dear community, there is relative index theorem due to Gromov and Lawson (Thm. 4.18 in POSITIVE SCALAR CURVATURE AND THE DIRAC OPERATOR ON COMPLETE RIEMANNIAN MANIFOLDS) which states that ...
4
votes
0answers
114 views

Element in spin group

I've got the following question: why is it true (if it really is?), that if I have a unitary element $u$ in the (real) Clifford algebra $Cl(V,g)$ which is even and the operator $\varphi(u)$ defined ...
4
votes
0answers
117 views

spectral sequence differential for cobordism

From page 6 of these solutions: the differential\begin{equation}d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})\end{equation}connecting the 1-st and the 2-nd row is the ...
4
votes
0answers
134 views

Duality between K-theory and K-homology in the non-compact, spin$^c$ case

Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong ...
1
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0answers
44 views

Norm of the operator acting on spinor bundle

Please forgive me if the question is too elementary, but however I was unable to manage by myself. The question comes from J.Varilly, H.Figueroa and J. Gracia-Bondia book "Elements of noncommutative ...
1
vote
0answers
101 views

Orthogonal trasformations with trivial spinor norm as product of reflections $r_w$ with $(w,w)=-2$

I'm trying to prove that, for a standard unimodular even lattice $\Lambda$ (by standard i mean that it is direct sum of copies of the hyperbolic plane $U$ and $E_8$) every element of $O^+(\Lambda)$ ...
1
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0answers
324 views

Low Dimensional Spin Manifolds

I am looking for examples of 2- and 3-dimensional flat spin manifolds with Euclidean and Lorentzian signatures, which admit parallel spinors and the dimension of the space of the parallel spinors is ...
1
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0answers
105 views

Topological index and Dirac operator with a non compact group

A spinor which belogs to a representation of a group $G=SO(p,q)$ is a section of a product bundle $S(M)\otimes E$, where $S(M)$ is a spin bundle over a four dimensional orientable and compact manifold ...
0
votes
0answers
62 views

Local behavior of Killing spinor on Sasaki-Einstein Manifold

I am trying to understand how a Killing spinor behaves near a closed Reeb orbit, for instance, on $S^5$ and $Y_{p,q}$ manifolds So Let us consider the Killing spinor equation on a five-dimensional ...
0
votes
0answers
29 views

Single-valueness of spinor components

I am confused about the non-single-valueness of spinor components. For instance, consider the Killing spinor $\psi$ on standard unit $S^3$: \begin{equation} {\nabla _m}\psi = - ...