## A characterization of the second Veronese embedding into a complex projective space.(English)Zbl 1038.53058

A smooth curve $$\gamma$$ on a Riemannian manifold $$M$$ is said to be of order 2 if it is parametrized by its arclength and if $$\dot{\gamma}, \nabla_{\dot{\gamma}}{\dot{\gamma}}$$ and $$\nabla_{\dot{\gamma}}\nabla_{\dot{\gamma}}{\dot{\gamma}}$$ are linear dependent at each point on the curve.
Assume now that $$M$$ is an $$n$$-dimensional Kähler manifold and let $$f:M \to {\mathbb{C}}P^ N(c)$$ be a Kähler isometric full immersion. If for each geodesic $$\gamma$$ on $$M$$ the image curve $$f \circ \gamma$$ on $${\mathbb{C}}P^ N(c)$$ is of order 2, then f is either a local congruence or $$f$$ is locally equivalent to the second Veronese embedding $$f_ 2:{\mathbb{C}}P^ n({c\over 2}) \to {\mathbb{C}}P^ N(c), N = {n(n+3)\over 2}$$.
This characterization theorem is a common improvement of results by K. Nomizu [Nagoya Math. J. 60, 181–188 (1976; Zbl 0305.53046)] and J. S. Pak and K. Sakamoto [Tôhoku Math. J. (2) 38, 297–311 (1986; Zbl 0602.53036)], who required that all geodesics are mapped to circles or that the image of each geodesic is locally contained in a totally real totally geodesic submanifold $${\mathbb{R}}P^ 2({c \over 4})$$ of $${\mathbb{C}}P^ N(c)$$, respectively.

### MSC:

 53C40 Global submanifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds

### Citations:

Zbl 0305.53046; Zbl 0602.53036
Full Text:

### References:

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