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Complex Structures on the Tangent Bundle of Riemannian Manifolds

  • Làszló Lempert
Part of the The University Series in Mathematics book series (USMA)

Abstract

It is well known that any (paracompact) differentiable manifold M has a complexification, i.e., a complex manifold XM, dimc X = dim M, such that M is totally real in X (see Ref. 8). It is also known that a small neighborhood U of M in X is diffeomorphic to the tangent bundle TM of M. Thus, the tangent bundle TM of any differentiable manifold carries a complex manifold structure. This complex structure is, of course, not unique. One way of finding a “canonical” complex structure is to endow M with some extra structure and require that the complex structure on TM interact with the structure of M. Here we consider smooth (meaning infinitely differentiable) Riemannian manifolds M. When M = ℝ, there is a natural identification Tℝ ≅ ℂ given by
$${{T}_{\sigma }}\mathbb{R} \mathrel\backepsilon \tau \frac{\partial }{{\partial \sigma }} \leftrightarrow \sigma + i\tau \in \mathbb{C},$$
(1.1)
and this endows Tℝ with a complex structure. In (1.1) σ denotes the coordinate on R. This coordinate depends on the algebraic structure of the identification (1.1); however, the complex structure on Tℝ depends only on the metric of ℝ. In other words, an isometry of ℝ induces a biholomorphic mapping on Tℝ.

Keywords

Riemannian Manifold Complex Manifold Tangent Bundle Conjugate Point Complete Riemannian Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Làszló Lempert
    • 1
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

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