A fundamental differential system of 3-dimensional Riemannian geometry

We briefly recall a fundamental exterior differential system introduced by the author and then apply it to the case of three dimensions. Here we find new global tensors and intrinsic invariants of oriented Riemaniann 3-manifolds. The system leads to a remarkable Weingarten type equation for surfaces on hyperbolic 3-space. An independent proof for low dimensions of the structural equations gives new insight on the intrinsic exterior differential system.

On vector bundle manifolds with spherically symmetric metrics

We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle $Erightarrow M$, over a Riemannian manifold $M$, when $E$ is endowed with a metric connection. The tangent bundle of $E$ admits a canonical decomposition and thus it is possible to define an interesting class of two-weights metrics with the weight functions depending on the fibre norm of $E$; hence the generalized concept of spherically symmetric metrics. We study its main properties and curvature equations. Finally we focus on a few applications and compute the holonomy of Bryant-Salamon type $mathrm{G}_2$ manifolds.