On a smooth manifold $M$ we introduce the concept of Codazzi-equivalent Riemannian metrics. The curvature tensors of two Codazzi-equivalent metrics satisfy a simple relation. The results together with known facts about Codazzi tensors give a method of proof for old and new local and global uniqueness results for Riemannian manifolds and Euclidean hypersurfaces.

We investigate genericities of reticular Lagrangian maps and reticular Legendrian maps in order to give generic classifications of caustics and wavefronts generated by a hypersurface germ without or with a boundary in a smooth manifold. We also give simpler proofs of main results in T. Tsukada, "Reticular Lagrangian Singularities," Asian J. Math., 1 (1997), pp. 572–622 and T. Tsukada, "Reticular Legendrian Singularities," Asian J. Math., 5:1 (2001), pp. 109–127.

We determine the action of the Torelli group on the equivariant cohomology of the space of flat $SL(2, C)$ connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even component of the space of flat $PSL(2, C)$ connections. The non-trivial part consists of the even alternating products of degree two Prym representations, so that the kernel of the action is precisely the Prym-Torelli group. We compute the Betti numbers of the ordinary cohomology of the moduli space of flat $SL(2, C)$ connections. Using results of Cappell-Lee-Miller we show that the Prym-Torelli group, which acts trivially on equivariant cohomology, acts non-trivially on ordinary cohomology.