Search

We describe the second order obstruction to deformation for nearly $G_2$ structures on compact manifolds. Building on work of B.Alexandrov and U.Semmelmann this allows proving rigidity under deformation for the proper nearly $G_2$ structure on the Aloff-Wallach space $N(1,1)$., Improved presentation,added full details on rigidity of Aloff-Wallach space $N(1,1)$. 18 pages

We show that a strict, nearly Kähler 6-manifold with either second or third Betti number nonzero is linearly unstable with respect to the $$\nu $$ν-entropy of Perelman and hence is dynamically unstable for the Ricci flow.

International audience; We characterise those complete Kähler manifolds supporting a nonconstant real-valued function with critical points whose Hessian is nonnegative, complex linear, has pointwise two eigenvalues and whose gradient is a Hessian-eigenvector.

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly K\"ahler or nearly parallel $\mathrm{G}_2$-manifolds, Sasakian and $3$-Sasakian manifolds, or twistor spaces over quaternion-K\"ahler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion. On every such manifold one can define a natural splitting of the tangent bundle which gives rise to a Riemannian submersion over a geometry with parallel skew-symmetric torsion of smaller dimension endowed with some extra structure. We show how previously known examples of geometries with parallel skew-symmetric torsion fit into this pattern, and construct several new examples. In the particular case where the above Riemannian submersion has the structure of a principal bundle, we give the complete local classification of the corresponding geometries with parallel skew-symmetric torsion., Comment: 42 pages; thoroughly revised version, including a simpler definition of the geometry with parallel curvature determined by a geometry with parallel skew-symmetric torsion, and an appendix discussing 3-(\alpha,\delta)-Sasakian structures in our framework

We show that Killing tensors on conformally flat $n$-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in the momenta., 8 pages

We introduce an appropriate formalism in order to study conformal Killing (symmetric) tensors on Riemannian manifolds. We reprove in a simple way some known results in the field and obtain several new results, like the classification of conformal Killing $2$-tensors on Riemannian products of compact manifolds, Weitzenb\"ock formulas leading to non-existence results, and construct various examples of manifolds with conformal Killing tensors., Comment: 29 pages; the statement and the proof of Theorem 5.1 have been corrected

We study generalized Killing spinors on round spheres $\mathbb{S}^n$. We show that on the standard sphere $\mathbb{S}^8$ any generalized Killing spinor has to be an ordinary Killing spinor. Moreover we classify generalized Killing spinors on $\mathbb{S}^n$ whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola--Friedrich's canonical spinor on 3-Sasakian manifolds of dimension 7. Finally we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors., 16 pages; new version filling a gap in the proof of Lemma 4.1 which was brought to our attention by Stanislav Wiechmann

We study generalized Killing spinors on compact Einstein manifolds with positive scalar curvature. This problem is related to the existence compact Einstein hypersurfaces in manifolds with parallel spinors, or equivalently, in Riemannian products of flat spaces, Calabi-Yau, hyperkaehler, G_2 and Spin(7) manifolds., 17 pages; new version including the 2-dimensional case and several stylistical improvements

An almost quaternion-Hermitian structure on a Riemannian manifold $(M^{4n},g)$ is a reduction of the structure group of $M$ to $\mathrm{Sp}(n)\mathrm{Sp}(1)\subset \mathrm{SO}(4n)$. In this paper we show that a compact simply connected homogeneous almost quaternion-Hermitian manifold of non-vanishing Euler characteristic is either a Wolf space, or $\mathbb{S}^2\times \mathbb{S}^2$, or the complex quadric $\mathrm{SO}(7)/\mathrm{U}(3)$., published version, references updated

We give criteria for real, complex and quaternionic representations to define $s$-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations whose second exterior power is irreducible or has an irreducible summand of co-dimension one, and we give a conceptual computation-free argument for the construction of the exceptional Lie algebras of compact type., Oberwolfach Preprints;2012,03

We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank homogeneous space has weakly complex tangent bundle if and only if it is a product of compact equal rank homogeneous spaces which either carry an invariant almost complex structure (and are classified by Hermann), or have stably trivial tangent bundle (and are classified by Singhof and Wemmer), or belong to an explicit list of weakly complex spaces which have neither stably trivial tangent bundle, nor carry invariant almost complex structures., 16 pages

The moduli space NK of infinitesimal deformations of a nearly K\"ahler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6-dimensional homogeneous nearly K\"ahler manifolds. It turns out that the nearly K\"ahler structure is rigid except for the flag manifold F(1,2)=SU_3/T^2, which carries an 8-dimensional moduli space of infinitesimal nearly K\"ahler deformations, modeled on the Lie algebra su_3 of the isometry group., Comment: 23 pages

In this article we give a unified treatment of the construction of all possible Weitzenb\"ock formulas for all irreducible, non--symmetric holonomy groups. The resulting classification is two--fold, we construct explicitly a basis of the space of Weitzenb\"ock formulas on the one hand and characterize Weitzenb\"ock formulas as eigenvectors for an explicitly known matrix on the other. Both classifications allow us to find tailor--suit Weitzenb\"ock formulas for applications like eigenvalue estimates or Betti number estimates., Comment: 48 pages

We study 6-dimensional nearly Kähler 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.

Conformal Killing forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We show the existence of conformal Killing forms on nearly Kaehler and weak G_2-manifolds. Moreover, we give a complete description of special conformal Killing forms. A further result is a sharp upper bound on the dimension of the space of conformal Killing forms., 24 pages

A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M_0,g_0). Extending the metric g_0 to a field g of bilinear forms g(p) on T_p M, p\in M_0, the pseudo Riemannian supergeometry of (M,g) is formulated as G-structure on M, where G is a supergroup with even part G_0\cong Spin(k,l); (k,l) the signature of (M_0,g_0). 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 X_s on M. Our main result is that X_s 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 X_s., 14 pages, latex, one typo corrected

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kaehler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms., Comment: published version