Universität zu Köln

Ostermayr, Dominik (2017). Some results in supergeometry: Harmonic maps from super Riemann surfaces and Automorphism supergroups of supermanifolds. PhD thesis, Universität zu Köln.

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Abstract

This thesis consists of two independent and self-contained parts. The first part is concerned with harmonic maps form super Riemann surfaces in complex projective spaces and projective spaces associated with the super skew-field $\mathbb{D}.$ In both cases, we develop the theory of Gau\ss\ transforms and study the notion of isotropy, in particular its relation to holomorphic differentials on the super Riemann surface. Moreover, we give a definition of finite type harmonic maps for a special class of maps into $\mathbb{C} P^{n|n+1}$ and thus obtain a classification for certain harmonic super tori. Furthermore, we investigate the equations satisfied by the underlying objects and give an example of a harmonic super torus in $\mathbb{D} P^2$ whose underlying map is not harmonic. In the second part, we study a classical theorem stating that the group of automorphisms of a manifold $M$ preserving a $G$-structure of finite type is a Lie group in the context of supermanifolds. We generalize this statement to the category of $cs$ manifolds and give some examples, some of which being generalizations of classical notions, others being particular to the super case. Notably, we have to introduce a new class of supermanifolds which we call mixed supermanifolds.