Marie-Amélie Lawn

AFFINE HYPERSPHERES ASSOCIATED TO SPECIAL PARA-KÄHLER MANIFOLDS

We prove that any special para-Kahler manifold is intrinsically an improper affine hypersphere. As a corollary, any para-holomorphic function F of n para-complex variables satisfying a non-degeneracy condition defines an improper affine hypersphere, which is the graph of a real function f of 2n variables. We give an explicit formula for the function f in terms of the para-holomorphic function F. Necessary and sufficient conditions for an affine hypersphere to admit the structure of a special para-Kahler manifold are given. Finally, it is shown that conical special para-Kahler manifolds are foliated by proper affine hyperspheres of constant mean curvature.

Decompositions of para-complex vector bundles and para-complex affine immersions

In this work we study decompositions of para-complex and para-holomorphic vector-bundles endowed with a connection ∇ over a para-complex manifold. First we obtain results on the connections induced on the subbundles, their second fundamental forms and their curvature tensors. In particular we analyze para-holomorphic decompositions. Then we introduce the notion of para-complex affine immersions and apply the above results to obtain existence and uniqueness theorems for para-complex affine immersions. This is a generalization of the results obtained by Abe and Kurosu [AK] to para-complex geometry. Further we prove that any connection with vanishing (0, 2)-curvature, with respect to the grading defined by the para-complex structure, induces a unique para-holomorphic structure.

Spinorial Characterizations of Surfaces into 3-dimensional Pseudo-Riemannian Space Forms

We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. This generalizes a recent work of the first author for spacelike immersed Lorentzian surfaces in ℝ2,1 to other Lorentzian space forms. We also characterize immersions of Riemannian surfaces in these spaces. From this we can deduce analogous results for timelike immersions of Lorentzian surfaces in space forms of corresponding signature, as well as for spacelike and timelike immersions of surfaces of signature (0, 2), hence achieving a complete spinorial description for this class of pseudo-Riemannian immersions.

A fundamental theorem for hypersurfaces in semi-Riemannian warped products

Abstract We find necessary and sufficient conditions for nondegenerate arbitrary signature manifolds to be realized as hypersurfaces in a large class of warped products manifolds. As an application, we give conditions for a 3-dimensional hypersurface in a 4-dimensional Robertson–Walker spacetime to be foliated by surfaces with lightlike or zero mean curvature and hence describe a way to study horizons in such spacetimes.

Spinorial representation of submanifolds in Riemannian space forms

In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Riemannian space forms in terms of the existence of so called generalized Killing spinors. We then discuss several applications, among them a new and concise proof of the fundamental theorem of submanifold theory. We also recover results of T. Friedrich, B. Morel and the authors in dimension 2 and 3.

A FUNDAMENTAL THEOREM FOR SUBMANIFOLDS OF MULTIPRODUCTS OF REAL SPACE FORMS

Abstract We prove a Bonnet theorem for isometric immersions of submanifolds into the products of an arbitrary number of simply connected real space forms. Then we prove the existence of associate families of minimal surfaces in such products. Finally, in the case of 𝕊2 × 𝕊2, we give a complex version of the main theorem in terms of the two canonical complex structures of 𝕊2 × 𝕊2.