The spin-geometry tag has no wiki summary.

**7**

votes

**0**answers

276 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**

votes

**0**answers

171 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

**0**answers

118 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

**0**answers

123 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

**0**answers

143 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 ...

**2**

votes

**0**answers

130 views

### Octonions product: inversion in the right and identity in the left

Once octonions product is studied, together with the relations with $Spin(8)$ and $SO(8)$ geometry (see for instance Robert Bryant's notes), one realises that the key fact bringing all the phenomena ...

**1**

vote

**0**answers

124 views

### Dirac operator in Generalized Geometry

I am wondering how the Dirac operator can be built in the context of Hichin's generalized geometry.
In particular, I have the following questions:
On a spin manifold, is the conventional spin ...

**1**

vote

**0**answers

91 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

**0**answers

103 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**

vote

**0**answers

329 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**

vote

**0**answers

110 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

**0**answers

70 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

**0**answers

30 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 = - ...