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.


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