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Pseudo-umbilical surfaces with constant Gauss curvature

Published online by Cambridge University Press:  20 January 2009

Bang-Yen Chen
Bang-Yen Chen
Affiliation:
Michigan State University, East Lansing, Michigan 48823
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Let M be a surface immersed in an m-dimensional space form Rm(c) of curvature c = 1, 0 or −1. Let h be the second fundamental form of this immersion; it is a certain symmetric bilinear mapping for x ∈ M, where Tx is the tangent space and the normal space of M at x. Let H be the mean curvature vector of M in Rm(c) and 〈, 〉 the scalar product on Rm(c). If there exists a function λ on M such that 〈h(X, Y), H〉 = λ〈X, Y〉 for all tangent vectors X, Y, then M is called a pseudo-umbilical surface of Rm(c). Let D denote the covariant differentiation of Rm(c) and η be a normal vector field. If we denote by D*η the normal component of Dη, then D* defines a connection in the normal bundle. A normal vector field η is said to be parallel in the normal bundle if Dη = 0. The length of mean curvature vector is called the mean curvature.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 18 , Issue 2 , December 1972 , pp. 143 - 148
Copyright
Copyright © Edinburgh Mathematical Society 1972

References

REFERENCES

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