CR Submanifolds of Hermitian Manifolds and the Tangential CR Equations
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 extensi
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CR Submanifolds of Hermitian Manifolds and the Tangential CR Equations Elisabetta Barletta and Sorin Dragomir
2010 MSC 32V10 · 32V25 · 53B25 · 53C40
4.1 Bejancu’s CR Submanifolds and CR Analysis The purpose of this (partially expository) paper is to set the basis for the mathematical analysis of solutions to the tangential Cauchy–Riemann equations ∂ M u = 0 on CR submanifolds M of Hermitian (e.g., Kählerian, locally conformal Kähler, etc.) manifolds M 2N , as introduced by A. Bejancu in his celebrated work [5], in an attempt to fill in a gap between the differential geometric side of the subject (e.g., devoted to the geometry of the second fundamental form of M in M 2N ) and the analysis problems related to the (local) properties of (weak) solutions to ∂ M u = 0 or the (local or global) holomorphic extension of CR functions on M.
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. E. Barletta (B) · S. Dragomir Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Via Dell’Ateneo Lucano 10, 85100 Potenza, Italy e-mail: [email protected] S. Dragomir e-mail: [email protected] © Springer Science+Business Media Singapore 2016 S. Dragomir et al. (eds.), Geometry of Cauchy–Riemann Submanifolds, DOI 10.1007/978-981-10-0916-7_4
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E. Barletta and S. Dragomir
4.1.1 CR Submanifolds Let M 2N be a Hermitian manifold, of complex dimension N, with the complex structure J and the Hermitian metric G. Let M be a real m-dimensional manifold, m = 2n + k < 2N, n ≥ 0, k ≥ 0, and : M → M 2N a C ∞ immersion. For every x ∈ M, the differential dx : Tx (M) → T(x) (M 2N ) has rank m hence (by the classical rank theorem) each point x0 ∈ M admits an open neighborhood x0 ∈ U ⊂ M such that : U → M 2n is an injective map. Therefore, through these notes, we shall assume that : M → M 2N is an injective immersion, rather than restrict the discussion to certain open sets (which would complicate the notation). Let bundle −1 T (M 2N ) → M be the pullback [the pullback of the tangent bundle T (M 2n ) → M 2n by ], i.e., −1 T (M 2n ) x = T(x) (M 2n ) for any x ∈ M. For each tangent vector field X ∈ X(M), we denote by ∗ X the section in the pullback bundle is given by (∗ X)(x) = (dx )Xx ∈ T(x) (M 2n ), x ∈ M. Let E() → M be the normal bundle of the given immersion, so that T(x) (M 2n ) = [(dx )Tx (M)] ⊕ E()x , x ∈ M. Let d be the codimension of M in M 2N , so that dimR E()x = d for any x ∈ M, and assume that d ≥ k. Let g = ∗ G be the first fundamental form of . Definition 1 A pair (M, D) consisting of a manifold M and a C ∞ distribution D of real rank 2n on M is a CR submanifold of type (n, k) of M 2n if (i) D is J-invariant, i.e., J(x) (dx ) Dx = (dx )Dx for any x ∈ M, and (ii) the orthogonal complement D⊥ of D in (T (M), g) is J-anti-invariant, i.e., J(x) (dx ) Dx⊥ ⊂ E()x for any x ∈ M. The inte
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