The structure of the main tensor of conformally connected torsion-free space. Conformal connections on hypersurfaces of projective space

We define a conformally connected space with arbitrary signature of angular metric and present basic formulas and classes of such spaces. We obtain the decomposition of the main tensor of a conformally connected torsion-free space into irreducible gauge-invariant summands and prove the following new property of the Weyl tensor: all affine connections obtained from the Levi-Civita connection via the normalization transformation have the same conformal Weyl tensor. We describe all conformal torsion-free connections on hypersurfaces of a projective space and give some examples. We construct a global conformal connection on a hyperquadric of the projective space.