Normal curvature of CR submanifolds of maximal CR dimension of the complex projective space
type: Item
instance of: scholarly article
title: Normal curvature of CR submanifolds of maximal CR dimension of the complex projective space
zbMATH Open document ID: 1413.53068
author: Erdal Karapınar
published in: Acta Mathematica Hungarica
publication date: 2019-02-01T00:00:00Z
review text: If \((\overline M,J)\) is a complex manifold of dimension \(\frac{n+p}{2}\) equipped with a Hermitian metric \(\overline g\), then an \(n\)-dimensional submanifold \(M\) is CR submanifold of maximal CR dimension if at each point \(x\in M\) the real dimension of \(JT_x(M)\cap T_x(M)\) is \(n-1\). The submanifold \(M\) is odd-dimensional and there exists a unit vector field \(\xi\) normal to \(TM\) such that \(JT_x(M)\subset T_x(M)\oplus\text{span}\{\xi_x\}\). If the normal curvature of \(M\) in \(\overline M\) vanishes identically, then it is said that the normal connection of \(M\) is flat. In this paper, the authors prove that there do not exist CR submanifolds \(M\) of maximal CR dimension of a complex projective space \(\mathbf{P(C)}\) of dimension \(\frac{n+p}{2}\) with flat normal connection \(D\) of \(M\) when the distinguished normal vector field \(\xi\) is parallel with respect to \(D\). For a submanifold \(M\) of \(\mathbf{P(C)}\), it is well known that \(\pi^{-1}(M)\) is a submanifold of \(\mathbf{S}^{n+p+1}\), where \(\pi^{-1}(M)\) is the circle bundle over \(M\)which is compatible with the Hopf map \(\pi:\mathbf{S}^{n+p+1}\to\mathbf{P(C)}\). If the normal connection of \(\pi^{-1}(M)\) in \(\mathbf{S}^{n+p+1}\) is flat, it is said that the normal connection of \(M\) is lift-flat. The authors also prove that if \(M\) is a CR submanifold of maximal CR dimension of a complex projective space \(\mathbf{P(C)}\) with lift-flat normal connection \(D\) and \(\xi\) is parallel with respect to \(D\), then there exists a totally geodesic complex projective subspace \(\mathbf{P}^{\frac{n+1}{2}}\mathbf{(C)}\) of \(\mathbf{P(C)}\) such that \(M\) is a real hypersurface of \(\mathbf{P}^{\frac{n+1}{2}}\mathbf{(C)}\).
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 7011025
zbMATH Keywords: CR submanifold
zbMATH Keywords: complex projective space
zbMATH Keywords: normal curvature
zbMATH Keywords: flat and lift-flat normal connection
MaRDI profile type: Publication
full work available at URL: s10474-018-0821-z
OpenAlex ID: W2801249403
Wikidata QID: Q125800431
DOI: 10.1007/S10474-018-0821-Z
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description: scientific article; zbMATH DE number 7011025