arXiv subjects
Publications and source records attributed to Kaveh Eftekharinasab.
In this paper we prove an infinite-dimensional version of Sard's theorem for Fréchet manifolds. Let $ M $ and $ N $ be bounded Fréchet manifolds such that the topologies of their model Fréchet spaces are defined by metrics with absolutely convex balls. Let $ f: M \rightarrow N $ be an $ MC^k$-Lipschitz-Fredholm map with $ k > \max \lbrace {\Ind f,0} \rbrace $. Then the set of regular values of $ f $ is residual in $ N $.
We obtained that any 2-form and any smooth function on 2-manifolds with boundary can be realized as the curvature form and the gaussian curvature function of some Riemmanian metric, respectively.
It was proved that the fundamental group of the space of harmonic polynomials of degree $n(n \geq 2)$, with the same Gaussian curvature is not trivial. Furthermore, we give an example of topologically nonequivalent conjugate harmonic functions having the same Gaussian curvature.