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Slant immersions. (English) Zbl 0677.53060
Let f: \(N\to M\) be an isometric immersion from a Riemannian manifold N into an almost Hermitian manifold M. For each non-zero tangent vector X in \(T_ pN\) at \(p\in N\) the Wirtinger angle \(\theta (X)\in [0,\pi /2]\) is defined as the angle between \(Jf_*X\) and \(f_*T_ pN\), where J is the almost complex structure on M. If \(\theta\) (X) is of constant value \(\theta\neq 0\) for all non-zero tangent vectors X to N, then f is called a slant immersion \((\theta =0\) characterizes the holomorphic and anti- holomorphic immersions into M). In case \(\theta =\pi /2\) 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 \({\mathbb{C}}^ 2\). Besides some characterizations of such surfaces he gives several (non-trivial) examples and obtains a classification of slant surfaces in \({\mathbb{C}}^ 2\) with parallel mean curvature vector. [Reviewer’s remark: Recently, the author and Y. Tazawa proved that every compact slant submanifold in \({\mathbb{C}}^ m\) is totally real [Slant submanifolds in complex number spaces (preprint)].]
Reviewer: J.Berndt

53C40 Global submanifolds
53A05 Surfaces in Euclidean and related spaces
Full Text: DOI
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