Higher order relations existing in normal coordinates between affine extensions of the curvature tensor and basic objects for any Fedosov supermanifolds are derived. Representation of these relations in general coordinates is discussed.
Generalizations of symplectic and metric structures for supermanifolds are analyzed. Two types of structures are possible according to the even/odd character of the corresponding quadratic tensors. In the even case one has a very rich set of geometri
c structures: even symplectic supermanifolds (or, equivalently, supermanifolds with non-degenerate Poisson structures), even Fedosov supermanifolds and even Riemannian supermanifolds. The existence of relations among those structures is analyzed in some details. In the odd case, we show that odd Riemannian and Fedosov supermanifolds are characterized by a scalar curvature tensor. However, odd Riemannian supermanifolds can only have constant curvature.
Extension of symplectic geometry on manifolds to the supersymmetric case is considered. In the even case it leads to the even symplectic geometry (or, equivalently, to the geometry on supermanifolds endowed with a non-degenerate Poisson bracket) or t
o the geometry on an even Fedosov supermanifolds. It is proven that in the odd case there are two different scalar symplectic structures (namely, an odd closed differential 2-form and the antibracket) which can be used for construction of symplectic geometries on supermanifolds.
We prove that a Kahler supermetric on a supermanifold with one complex fermionic dimension admits a super Ricci-flat supermetric if and only if the bosonic metric has vanishing scalar curvature. As a corollary, it follows that Yaus theorem does not hold for supermanifolds.
We present a higher order generalisation of the clockwork mechanism starting from an underlying non-linear multigravity theory with a single scale and nearest neighbour ghost-free interactions. Without introducing any hierarchies in the underlying po
tential, this admits a family of Minkowski vacua around which massless graviton fluctuations couple to matter exponentially more weakly than the heavy modes. Although multi-diffeomorphisms are broken to the diagonal subgroup in our theory, an asymmetric distribution of conformal factors in the background vacua translates this diagonal symmetry into an asymmetric shift of the graviton gears. In particular we present a TeV scale multigravity model with ${cal O}(10)$ sites that contains a massless mode whose coupling to matter is Planckian, and a tower of massive modes starting at a TeV mass range and with TeV strength couplings. This suggests a possible application to the hierarchy problem as well as a candidate for dark matter.