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http://cimat.repositorioinstitucional.mx/jspui/handle/1008/712| CR Structures and Partially Pure Spinors | |
| IVAN TELLEZ TELLEZ | |
| Acceso Abierto | |
| Atribución-NoComercial | |
| Spin Geometry CR structures | |
| The main subject of this work is the study of CR manifolds via twisted spin geometry. Our main motivation is the relationship between classical pure spinors and orthogonal almost complex structures. More precisely, a classical pure spinor $phiinDelta_{2m}$ is a spinor such that the (isotropic) subspace of complexified vectors $X-iYinmathbb{R}^{2m}otimes mathbb{C}$, $X,Yinmathbb{R}^{2m}$, which annihilate $phi$ under Clifford multiplication [(X-iY)cdot phi =0] is of maximal dimension, where $minmathbb{N}$ and $Delta_{2m}$ is the standard complex representation of the Spin group $Spin(2m)$. This means that for every $Xinmathbb{R}^{2m}$ there exists a $Yinmathbb{R}^{2m}$ satisfying [Xcdot phi = i Ycdot phi. ] By setting $Y=J(X)$, one can see that a pure spinor determines a complex structure on $mathbb{R}^{2m}$. Geometrically, the two subspaces $TMcdotphi$ and $i,TMcdotphi$ of $Delta_{2m}$ coincide, which means $TMcdot phi$ is a complex subspace of $Delta_{2m}$, and the effect of multiplication by the number $i=sqrt{-1}$ is transferred to the tangent space $TM$ in the form of $J$. Authors like Charlton and Trautman have investigated (the classification of) non-pure classical spinors by means of their isotropic subspaces. Trautman noted that there may be many spinors (in different orbits under the action of the Spin group) admitting isotropic subspaces of the same dimension, and that there is a gap in the possible dimensions of such isotropic subspaces. Following these geometrical considerations, we investigated how to define twisted partially pure spinors in order to spinorially characterize subspaces of Euclidean space endowed with a complex structure. We characterize subspaces of Euclidean space $mathbb{R}^{n}$, $ngeq 2m$, endowed with an orthogonal complex structure by means of twisted spinors. Thus, we define twisted partially pure spinors (cf. Definition 2.2.1) in order to establish a one-to-one correspondence between subspaces of Euclidean space (of fixed codimension and endowed with a orthogonal complex structures and oriented orthogonal complements), and orbits of such spinors under a particular subgroup of the twisted spin group (cf. Theorem 2.2.1). By using spinorial twists we avoid having different orbits under the full twisted spin group and also the aforementioned gap in the dimensions. The need to establish such a correspondence arises from our interest in developing a spinorial setup to study the geometry of manifolds admitting | |
| 01-10-2017 | |
| Tesis de doctorado | |
| Inglés | |
| PROBABILIDAD | |
| Versión aceptada | |
| acceptedVersion - Versión aceptada | |
| Aparece en las colecciones: | Tesis del CIMAT |
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