Surfaces of Orthonormal Frame Bundle with Wagner Lift Metric

Mikhail Malakhaltesev

Universidad de los Andes (Colombia)

Abstract:

The talk is based on the paper E. S. Becerra, M. Malakhaltsev, and Haimer A. Trejos, "Surfaces of Orthonormal Frame Bundle with Wagner Lift Metric", Lobachevskii Journal of Mathematics, 2022, Vol. 43, No. 4, pp. 785–798. For a two-dimensional oriented Riemannian manifold  \[(M,g)\] , we consider the horizontal and vertical surfaces immersed in the 3-dimensional Riemannian manifold  \[SO(M,g)\] , the total space of positive orthonormal frame bundle over  \[(M,g)\] , endowed by the Wagner lift metric  \[G\] .

The horizontal surfaces are the sections of  \[SO(M,g)\] , and the vertical surfaces are the preimages of regular curves of  \[M\] . We find the first and second fundamental form of the horizontal and vertical surfaces, and the mean curvature of these surfaces. Using these results, we exhibit suitable conditions for the existence of minimal surfaces and constant mean curvature surfaces in  \[(SO(M,g),G)\] .