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Complex Structures on the Tangent Bundle of Riemannian Manifolds

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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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© 1993 Springer Science+Business Media New York

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Lempert, L. (1993). Complex Structures on the Tangent Bundle of Riemannian Manifolds. In: Ancona, V., Silva, A. (eds) Complex Analysis and Geometry. The University Series in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-9771-8_8

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  • DOI: https://doi.org/10.1007/978-1-4757-9771-8_8

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