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- Title
C-R Immersions and Sub-Riemannian Geometry.
- Authors
Barletta, Elisabetta; Dragomir, Sorin; Esposito, Francesco
- Abstract
On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form θ , we consider natural ϵ -contractions, i.e., contractions g ϵ M of the Levi form G θ , such that the norm of the Reeb vector field T of (M , θ) is of order O (ϵ − 1) . We study isopseudohermitian (i.e., f ∗ Θ = θ ) Cauchy–Riemann immersions f : M → (A , Θ) between strictly pseudoconvex CR manifolds M and A, where Θ is a contact form on A. For every contraction g ϵ A of the Levi form G Θ , we write the embedding equations for the immersion f : M → A , g ϵ A . A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as ϵ → 0 + . For every isopseudohermitian immersion f : M → S 2 N + 1 into a sphere S 2 N + 1 ⊂ C N + 1 , we show that Webster's pseudohermitian scalar curvature R of (M , θ) satisfies the inequality R ≤ 2 n (f ∗ g Θ) (T , T) + n + 1 + 1 2 { ∥ H (f) ∥ g Θ f 2 + ∥ trace G θ Π H (M) ∇ ⊤ − ∇ ∥ f ∗ g Θ 2 } with equality if and only if B (f) = 0 and ∇ ⊤ = ∇ on H (M) ⊗ H (M) . This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms.
- Subjects
MATTHEW, the Apostle, Saint; VECTOR fields; GEOMETRY; PSEUDOCONVEX domains; CURVATURE; RIEMANNIAN geometry
- Publication
Axioms (2075-1680), 2023, Vol 12, Issue 4, p329
- ISSN
2075-1680
- Publication type
Article
- DOI
10.3390/axioms12040329