Abstract
We exhibit the relationship between the second fundamental form and the Levi form of a CR submanifold M (in the sense of A. Bejancu, [5]) in a Hermitian (e.g., Kählerian or locally conformal Kähler) manifold \(M^{2N}\) and start a study of the CR extension problem from M to \(M^{2N}\).
Dedicated to Aurel Bejancu—A. Bejancu (B.): Romanian mathematician (b. 1946). The scientific creation of B. is mainly devoted to the theory of isometric immersions in (semi) Riemannian and Finslerian geometry.
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Notes
- 1.
The term is often reserved for CR submanifolds of \({\mathbb C}^N\) for some \(N \ge 2\).
- 2.
- 3.
A CR immersion is a \(C^\infty \) immersion and a CR map.
- 4.
CR immersions are CR analogs to holomorphic immersions and any isometric holomorphic immersion into a Kählerian manifold is known to be minimal (has vanishing mean curvature).
- 5.
The particular case \(\xi _{\mathcal D} = 0\) (equivalently \(\xi \in {\mathcal D}^\bot \)) may be treated in a similar manner.
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Acknowledgments
The authors are grateful to M. Hasan Shahid (Jamia Millia Islamia University) for inviting them to contribute to the present volume. They also acknowledge support from P.R.I.N. 2012 (Italy).
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Barletta, E., Dragomir, S. (2016). CR Submanifolds of Hermitian Manifolds and the Tangential CR Equations. In: Dragomir, S., Shahid, M., Al-Solamy, F. (eds) Geometry of Cauchy-Riemann Submanifolds. Springer, Singapore. https://doi.org/10.1007/978-981-10-0916-7_4
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