Parallel submanifolds of complex projective space and their normal holonomy |
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Authors: | Sergio Console Antonio J Di Scala |
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Institution: | (1) Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy;(2) Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy |
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Abstract: | The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the
complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal
holonomy approach. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the (projectivized)
isotropy representation of an irreducible Hermitian symmetric space. Moreover, we show how these important submanifolds are
related to other areas of mathematics and theoretical physics. Finally, we state a conjecture about the normal holonomy group
of a complete and full complex submanifold of the complex projective space.
Research partially supported by GNSAGA (INdAM) and MIUR of Italy. |
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Keywords: | Normal holonomy group Symmetric submanifolds Parallel second fundamental form Normal bundle |
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