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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

##### MSC:
 53C40 Global submanifolds 53A05 Surfaces in Euclidean and related spaces
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##### References:
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