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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The curvatures of some skew fundamental forms

Author: Tilla Klotz Milnor
Journal: Proc. Amer. Math. Soc. 62 (1977), 323-329
MSC: Primary 53C40
MathSciNet review: 0461383
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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'$I$ ' =$   II$ - H$I$ , H'$II$ ' = H$II$ - K$I and III$ ' = H$II$ ' - K$I$ '$, 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.

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Article copyright: © Copyright 1977 American Mathematical Society