Uwe Semmelmann

On nearly parallel G2-structures

Abstract A nearly parallel G 2 -structure on a seven-dimensional Riemannian manifold is equivalent to a spin structure with a Killing spinor. We prove general results about the automorphism group of such structures and we construct new examples. We classify all nearly parallel G 2 -manifolds with large symmetry group and in particular all homogeneous nearly parallel G 2 -structures.

Eigenvalue estimates for the Dirac operator on quaternionic Kähler manifolds

Abstract. We consider the Dirac operator on compact quaternionic Kähler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space.

Killing spinors are Killing vector fields in Riemannian Supergeometry

A supermanifold M is canonically associated to any pseudo-Riemannian spin manifold (M0, g0). Extending the metric g0 to a field g of bilinear forms g(p) on TpM, p ϵ M0, the pseudo-Riemannian supergeometry of (M, g) is formulated as G-structure on M, where G is a supergroup with even part G0 ≊ Spin(k, l); (k, l) the signature of (M0, go). Killing vector fields on (M, g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field Xs on M. Our main result is that Xs is a Killing vector field on (M, g) if and only if s is a twistor spinor. In particular, any Killing spinor s defines a Killing vector field Xs.

Parallel spinors and holonomy groups

In this paper we complete the classification of spin manifolds admitting parallel spinors, in terms of the Riemannian holonomy groups. More precisely, we show that on a given n-dimensional Riemannian manifold, spin structures with parallel spinors are in one to one correspondence with lifts to Spinn of the Riemannian holonomy group, with fixed points on the spin representation space. In particular, we obtain the first examples of compact manifolds with two different spin structures carrying parallel spinors.

CLIFFORD STRUCTURES ON RIEMANNIAN MANIFOLDS

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r = 2 and r = 3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kahler, quaternion-Kahler and Riemannian products of quaternion-Kahler manifolds for r = 2, 3 and 4 respectively, several classes of 8-dimensional manifolds (for 5 � r � 8), families of real, complex and quaternionic Grassmannians (for r = 8, 6 and 5 respectively), and Rosenfelds elliptic projective planes OP 2 , (C O)P 2 , (H O)P 2 and (O O)P 2 , which are symmetric spaces associated to the exceptional simple Lie groups F4, E6, E7 and E8 (for r = 9, 10, 12 and 16 respectively). As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry. 2000 Mathematics Subject Classification: Primary 53C26, 53C35, 53C10, 53C15.

UNIT KILLING VECTOR FIELDS ON NEARLY KÄHLER MANIFOLDS

We study 6-dimensional nearly Kahler manifolds admitting a Killing vector field of unit length. In the compact case, it is shown that up to a finite cover there is only one geometry possible, that of the 3-symmetric space S3 × S3.

The First Eigenvalue of the Dirac Operator on Quaternionic Kähler Manifolds

In a previous paper we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kaehler manifolds. In the present article we show that the only manifolds in the limit case, i.e. the only manifolds where the lower bound is attained as an eigenvalue, are the quaternionic projective spaces. We use the equivalent formulation in terms of the quaternionic Killing equation and show that a nontrivial solution defines a parallel spinor on the associated hyperkaehler manifold.

Complex Contact Structures and the First Eigenvalue of the Dirac Operator on Kähler Manifolds.

In this paper Kählerian Killing spinors on manifolds of complex dimensionm=4l+3 are constructed. The construction is based on a theorem which states that a closed Kähler Einstein manifold of complex dimension 4l+3 and positive scalar curvature admits a Kählerian Killing spinor if and only if there is a complex (2l+1)-contact structure. In particular, any complex contact structure in the usual sense gives rise to such a generalized contact structure. Using this, new examples of Kählerian Killing spinors are obtained.

The Weitzenböck machine

Weitzenbock formulas are an important tool in relating local differential geometry to global topological properties by means of the so-called Bochner method. In this article we give a unified treatment of the construction of all possible Weitzenbock formulas for all irreducible, non-symmetric holonomy groups. We explicitly construct a basis of the space of Weitzenbock formulas. This classification allows us to find customized Weitzenbock formulas for applications such as eigenvalue estimates or Betti number estimates.

Killing Forms on Symmetric Spaces

Abstract Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We show that a compact simply connected symmetric space carries a non-parallel Killing p-form ( p ⩾ 2 ) if and only if it isometric to a Riemannian product S k × N , where S k is a round sphere and k > p .