The geometry of surfaces whose shape operator matrix along a surface curve is almost complex structure and almost product structure

Abstract

In this talk, the geometry of a surface is examined when the shape operator matrix of a surface along a surface curve is almost complex structure, almost product structure and almost tangent structure. First, it was shown that the shape operator matrix of a surface along a surface curve cannot be almost complex structure. Then, two cases were considered when the shape operator matrix of a surface along a surface curve is almost product structure. First, when the geodesic curvature of the surface is different from zero, the Gaussian curvature, mean curvature, umbilicality of the surface were examined and the principal curvatures were found. In the second case, when the geodesic curvature of the surface is zero, the character of the surface was examined. Finally, the Gaussian curvature, mean curvature and principal curvatures of the surface are determined in the case where the shape operator matrix is almost tangent structure.