Conformal Killing forms on Riemannian manifolds

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.Comment: 24 page

Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian manifolds

For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of $3$-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete $3$-Sasakian structure of dimension $4d+3$, with $d>1$, has sub-Riemannian diameter bounded by $\pi$. When $d=1$, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on $\mathbb{S}^{4d+3}$ of the quaternionic Hopf fibrations: \begin{equation*} \mathbb{S}^3 \hookrightarrow \mathbb{S}^{4d+3} \to \mathbb{HP}^d, \end{equation*} whose exact sub-Riemannian diameter is $\pi$, for all $d \geq 1$.Comment: 34 pages, v2: fixed and clarified the proof of Theorem 7 and some typos, v3: final version, to appear on Journal of the Institute of Mathematics of Jussie

Tri-Sasakian consistent reduction

We establish a universal consistent Kaluza-Klein truncation of M-theory based on seven-dimensional tri-Sasakian structure. The four-dimensional truncated theory is an N=4 gauged supergravity with three vector multiplets and a non-abelian gauge group, containing the compact factor SO(3). Consistency follows from the fact that our truncation takes exactly the same form as a left-invariant reduction on a specific coset manifold, and we show that the same holds for the various universal consistent truncations recently put forward in the literature. We describe how the global symmetry group SL(2,R) x SO(6,3) is embedded in the symmetry group E7(7) of maximally supersymmetric reductions, and make the connection with the approach of Exceptional Generalized Geometry. Vacuum AdS4 solutions spontaneously break the amount of supersymmetry from N=4 to N=3,1 or 0, and the spectrum contains massive modes. We find a subtruncation to minimal N=3 gauged supergravity as well as an N=1 subtruncation to the SO(3)-invariant sector. We also show that a reduction on the homogeneous space N^{010} enhances the universal tri-Sasakian truncation with a Betti vector multiplet.Comment: 40 pages main text, 9 pages appendix, 1 figure, 6 tables, v2: JHEP version, added references, minor corrections, changed notation fluctuations in tables 2-

Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.Comment: 49 page

Sasaki-Einstein Manifolds

This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.Comment: 58 pages. Invited contribution to Surveys in Differential Geometry. v2: references and discussion adde

On the infinitesimal isometries of manifolds with Killing spinors

International audienceWe study the Lie algebra of infinitesimal isometries of seven-dimensional simply connected manifolds with Killing spinors. We obtain some splitting theorems for the action of this algebra on the space of Killing spinors, and as a corollary we prove that there is no infinitesimal isometry of constant length on a seven-dimensional 3-Sasakian manifold (not isometric to a space form) except the linear combinations of the Sasakian vector fields

Spin(9) geometry of the octonionic Hopf fibration

We deal with Riemannian properties of the octonionic Hopf fibration S^{15}-->S^8, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S^1 subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.Comment: Proofs and Examples revised, some references adde