# Spinorial methods, para-complex and para-quaternionic geometry in the theory of submanifolds

Abstract : This thesis is devoted to the theory of immersions, using methods of spin geometry, para-complex and para-quaternionic geometry. It is subdivided into three different topics. The first two are related to the study of conformal immersions of pseudo-Riemannian surfaces. On the one hand we study the immersion into three-dimensional pseudo-Euclidean spaces: with the methods of para-complex geometry and using real spinor representations, we prove the equivalence between the data of a conformal immersion of a Lorentzian surface in $\mathbb{R}^{2,1}$ and spinors satisfying a Dirac-type equation. On the other hand, we consider immersions of such surfaces into the four-dimensional pseudo-sphere $\mathbb{S}^{2,2}$: a one-to-one correspondence between such immersions and para-quaternionic line subbundles of the trivial bundle $M\times\mathbb{H}^2$ is given. Considering a particular (para-)complex structure on this bundle, namely the mean curvature pseudo-sphere congruence, and the para-quaternionic Hopf fields of the immersion, we define the Willmore functional of the surface and can express its energy as the sum of this functional and of a topological invariant. The last topic is more general and deals with para-complex vector bundles and para-complex affine immersions. We introduce para-holomorphic vector bundles and characterize para-holomorphic subbundles and subbundles of type $(1,1)$ in terms of the associated induced connections and second fundamental forms. The fundamental equations for general decompositions of vector bundles with connection are studied in the case where some of the (sub)bundle are para-holomorphic in order to prove existence and uniqueness theorems of para-complex affine immersions.
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Cited literature [48 references]

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Submitted on : Friday, April 20, 2007 - 12:27:42 PM
Last modification on : Friday, July 9, 2021 - 11:30:39 AM
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• HAL Id : tel-01746544, version 2

### Citation

Marie-Amelie Lawn-Paillusseau. Spinorial methods, para-complex and para-quaternionic geometry in the theory of submanifolds. Mathematics [math]. Université Henri Poincaré - Nancy 1; Rheinische DEiedrich-Wilhelms-Universität Bonn, 2006. English. ⟨NNT : 2006NAN10182⟩. ⟨tel-01746544v2⟩

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