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On real local isometric immersions of \({\mathbb{C}Q^2_c}\) into \({\mathbb{C} P^{3}}\)

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Abstract

We investigate real local isometric immersions of Kähler manifolds \({\mathbb{C}Q^2_c}\) of constant holomorphic curvature 4c into complex projective 3-space. Our main result is that the standard embedding of \({\mathbb{C}P^2}\) into \({\mathbb{C}P^3}\) has strong rigidity under the class of local isometric transformations. We also prove that there are no local isometric immersions of \({\mathbb{C}Q^2_c}\) into \({\mathbb{C}P^3}\) when they have different holomorphic curvature. An important method used is a study of the relationship between the complex structure of any locally isometric immersed \({\mathbb{C}Q^2_c}\) and the complex structure of the ambient space \({\mathbb{C}P^3}\).

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  1. The Faculty of Informatics and e-Learning, Sør-Trøndelag University College, 7004, Trondheim, Norway

    Hans Jakob Rivertz

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  1. Hans Jakob Rivertz
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Correspondence to Hans Jakob Rivertz.

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In memoriam Per Tomter.

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Rivertz, H.J. On real local isometric immersions of \({\mathbb{C}Q^2_c}\) into \({\mathbb{C} P^{3}}\) . J. Geom. 104, 357–374 (2013). https://doi.org/10.1007/s00022-013-0157-3

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  • DOI: https://doi.org/10.1007/s00022-013-0157-3

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