The curvatures of some skew fundamental forms

Abstract:

Fix a unit normal vector field on a surface ${C^4}$-immersed in a Riemannian 3-manifold of constant sectional curvature. Suppose H and K are mean and Gauss curvatures respectively, and that $H’ = \sqrt {{H^2} - K}$. Wherever $H’ \ne 0$, define I’, II’ and III’ by $H’\text {I}’ = \text {II} - H\text {I}, H’\text {II}’ = H\text {II} - K\text {I}$ and $\text {III}’ = H\text {II}’ - K\text {I}’$, where I and II are the first and second fundamental forms. For constants $\alpha ,\beta$, and $\gamma$, let $\Lambda ’ = \alpha {\text {I}}’ + \beta {\text {II}}’ + \gamma {\text {III}}’$. Wherever $H’ \ne 0$ and $\Lambda ’$ is nondegenerate, the curvature of this (not necessarily Riemannian) metric $\Lambda ’$ is computed in terms of $K({\text {I}}’)$ and more familiar quantities on the surface. Some discussion of $K({\text {I}}’)$ is also included.