SEMINARIO DE GEOMETRIA

TÍTULO: Homogeneous Riemannian manifolds with nullity.

AUTOR: Carlos E. Olmos (Famaf, UNC/Ciem, CONICET).

AULA 320, Departamento de Matemáticas.

FECHA Y HORA: 11:30, viernes 11 de noviembre de 2022.

ABSTRACT: (ver fichero adjunto) We will speak about joint results with Antonio J. Di Scala and Francisco Vittone, about the structure of irreducible homogeneous Riemannian manifolds $M^n= G/H$ whose curvature tensor has non-trivial nullity. In a recent paper we developed a general theory to deal with such spaces. By making use of this theory we were able to construct the first non-trivial examples in any dimension. Such examples have the minimum possible conullity $k=3$.

The key fact is the existence of a non-trivial transvection $X$ at $p$ (i.e., Killing fields $(
abla X)_p = 0$) such that $X_p$ is not in the nullity subspace $
u_p$ at $p$ but the Jacobi operator $R_{cdot, X_p}X_p$ is zero. The nullity distribution $
u$ is highly non-homogeneous in the sense that no non-trivial Killing field lie in $
u$ and hence $
u$ is not given by the tangent spaces of orbits of an isometry subgroup of $G$. The Lie algebra $mathfrak g$ of $G$ is never reductive (in particular, $M$ is not compact).

After surveying on these results we will present recent results that give a substantial improvement for the structure of homogeneous spaces in relation with the nullity. By some rather delicate argument we showed that there is always a transvection, possibly enlarging the presentation group $G$, in any direction $
u _p$ of the nullity, for all $pin M$.

Moreover, such transvections form an abelian ideal $mathfrak a$ of $mathfrak g$ which implies, if $k=3$, that $G=mathbb R^{n-1}rtimes mathbb R$ and $H$ is trivial.

On the one hand, the leaves of the nullity are Euclidean spaces and the projection to the quotient space $M/
u$ is never a Riemannian submersion.

But on the other hand, the foliation given by the orbits of the normal subgroup $Asubset G$, associated to $mathfrak a $ is intrinsically flat, contains the nullity foliation and the projection to the quotient is a Riemannian submersion.

 

Finally, we will show some simply connected examples with non-trivial topology and compact quotients. This answers a natural question.