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Approximation of Regular Sasakian Manifolds

Giovanni Placini

Abstract

We investigate the problem of approximating a regular Sasakian structure by CR immersions in a standard sphere. Namely, we show that this is always possible for compact Sasakian manifolds. Moreover, we prove an approximation result for non-compact $η$-Einstein manifolds via immersions in the infinite dimensional sphere. We complement this with several examples.

Approximation of Regular Sasakian Manifolds

Abstract

We investigate the problem of approximating a regular Sasakian structure by CR immersions in a standard sphere. Namely, we show that this is always possible for compact Sasakian manifolds. Moreover, we prove an approximation result for non-compact -Einstein manifolds via immersions in the infinite dimensional sphere. We complement this with several examples.
Paper Structure (6 sections, 5 theorems, 38 equations)

This paper contains 6 sections, 5 theorems, 38 equations.

Table of Contents

  1. Introduction and statements of the main results
  2. Sasakian manifolds
  3. CR immersions of regular and complete Sasakian manifolds into spheres
  4. Approximation of compact regular structures via immersions into spheres
  5. Approximation of $\eta$-Einstein regular structures
  6. Explicit examples of approximations of $\eta$-Einstein structures

Key Result

Theorem 1

Let $(M,\eta,g)$ be a compact regular Sasakian manifold. Then there exist a sequence of CR immersions $\varphi_k:M\longrightarrow S^{2N+1}$ into standard Sasakian spheres such that suitable transverse homotheties of the induced structures converge to $(\eta, g)$ in the $C^\infty$-norm.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Boyer08Book
  • Definition 4: $\mathcal{D}$-homothety or a transverse homothety
  • Example 5: Standard Sasakian sphere
  • Definition 6: Transverse Kähler deformations
  • Definition 7: Sasakian immersion and embedding
  • Remark 8
  • Proposition 9: LoiPlaciniZedda23SasakiHomogeneous
  • Proposition 10
  • ...and 7 more