arXiv subjects
Publications and source records attributed to Lars Schäfer.
In this paper we provide a second variation formula for L-minimal Lagrangian submanifolds in a pseudo-Sasakian manifold. We apply it to the case of Lorentzian-Sasakian manifolds and relate the L-stability of L-minimal Legendrians in a Sasakian M to their L-stability in an associated Lorentzian-Sasakian structure on M.
In this paper we provide examples of maps from almost complex domains into pseudo-Riemannian symmetric targets, which are pluriharmonic and not integrable, i.e. do not admit an associated family. More precisely, for one class of examples the source has a non-integrable complex structure, like for instance a nearly Kaehler structure and the target is a Riemannian symmetric space and for the other class the source is a complex manifold and the target is a pseudo-Riemannian symmetric space. These examples show, that a former result on the existence of associated families is sharp.
Firstly we give a condition to split off the K"ahler factor from a nearly pseudo-K"ahler manifold and apply this to get a structure result in dimension 8. Secondly we extend the construction of nearly K"ahler manifolds from twistor spaces to negatively curved quaternionic K"ahler manifolds and para-quaternionic K"ahler manifolds. The class of nearly pseudo-K"ahler manifolds obtained from this construction is characterized by a holonomic condition. The combination of these results enables us to give a classification result in (real) dimension 10. Moreover, we show that a strict nearly pseudo-K"ahler six-manifold is Einstein.
The subject of this paper is six-dimensional nearly (para-)Kähler geometry with pseudo-Riemannian metrics. Firstly, we derive the analogue of the well-known exterior differential system characterising a nearly Kähler manifold and prove applications to the automorphism group of a nearly (para-)Kähler structure. Secondly, we prove existence and uniqueness results for left-invariant nearly (para-)Kähler structures on Lie groups $G \times G$ where $G$ is three-dimensional and simple.
Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element η\in Λ^3L\subset Λ^3V. If ηis of type (3,0)+(0,3) with respect to a skew-symmetric endomorphism J with J^2=\e Id, then the Lie group {\cal L}(η) is endowed with a left-invariant nearly Kähler structure if \e =-1 and with a left-invariant nearly para-Kähler structure if \e =+1. This construction exhausts all complete simply connected flat nearly (para-)Kähler manifolds. If η\neq 0 has rational coefficients with respect to some basis, then {\cal L}(η) admits a lattice Γ, and the quotient Γ\setminus {\cal L}(η) is a compact inhomogeneous nearly (para-)Kähler manifold. The first non-trivial example occurs in six dimensions.
It was proven by Hitchin that any solution of his evolution equations for a half-flat SU(3)-structure on a compact six-manifold M defines an extension of M to a seven-manifold with holonomy in G_2. We give a new proof, which does not require the compactness of M. More generally, we prove that the evolution of any half-flat G-structure on a six-manifold M defines an extension of M to a Ricci-flat seven-manifold N, for any real form G of SL(3,C). If G is noncompact, then the holonomy group of N is a subgroup of the noncompact form G_2^* of G_2^C. Similar results are obtained for the extension of nearly half-flat structures by nearly parallel G_2- or G_2^*-structures, as well as for the extension of cocalibrated G_2- and G_2^*-structures by parallel Spin(7)- and Spin(3,4)-structures, respectively. As an application, we obtain that any six-dimensional homogeneous manifold with an invariant half-flat structure admits a canonical extension to a seven-manifold with a parallel G_2- or G_2^*-structure. For the group H_3 \times H_3, where H_3 is the three-dimensional Heisenberg group, we describe all left-invariant half-flat structures and develop a method to explicitly determine the resulting parallel G_2- or G_2^*-structure without integrating. In particular, we construct three eight-parameter families of metrics with holonomy equal to G_2 and G_2^*. Moreover, we obtain a strong rigidity result for the metrics induced by a half-flat structure (ω,ρ) on H_3 \times H_3 satisfying ω(Z,Z)=0 where Z denotes the centre. Finally, we describe the special geometry of the space of stable three-forms satisfying a reality condition. Considering all possible reality conditions, we find four different special Kähler manifolds and one special para-Kähler manifold.
We show that Lagrangian submanifolds in six-dimensional nearly Kähler (non Kähler) manifolds and in twistor spaces $Z\sp{4n+2}$ over quaternionic Kähler manifolds $Q\sp{4n}$ are minimal. Moreover, we will prove that any Lagrangian submanifold $L$ in a nearly Kähler manifold $M$ splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly Kähler part of $M$ and the second factor is Lagrangian in the Kähler part of $M$. Using this splitting theorem we then describe Lagrangian submanifolds in nearly Kähler manifolds of dimensions six, eight and ten.
We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature $(2m,2m)$ with $m\ge 3$. Moreover, the geometry of the second factor is encoded in a complex three-form $ζ\in Λ^3 (\mathbb{C}^m)^*$. The first nontrivial example occurs in dimension $4m=12$.