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Slant immersions. (English) Zbl 0677.53060

Let f: NM be an isometric immersion from a Riemannian manifold N into an almost Hermitian manifold M. For each non-zero tangent vector X in T p N at pN the Wirtinger angle θ(X)[0,π/2] is defined as the angle between Jf * X and f * T p N, where J is the almost complex structure on M. If θ (X) is of constant value θ0 for all non-zero tangent vectors X to N, then f is called a slant immersion (θ=0 characterizes the holomorphic and anti- holomorphic immersions into M). In case θ=π/2 the immersion f is also called totally real.

At first the author proves some fundamental properties of slant immersions. Then he restricts to the special case of slant surfaces in the complex 2-plane 2 . Besides some characterizations of such surfaces he gives several (non-trivial) examples and obtains a classification of slant surfaces in 2 with parallel mean curvature vector. [Reviewer’s remark: Recently, the author and Y. Tazawa proved that every compact slant submanifold in m is totally real [Slant submanifolds in complex number spaces (preprint)].]

Reviewer: J.Berndt

MSC:
53C40Global submanifolds (differential geometry)
53A05Surfaces in Euclidean space