Complex Analysis and Geometry pp 235-251 | Cite as

# Complex Structures on the Tangent Bundle of Riemannian Manifolds

Chapter

## Abstract

It is well known that any (paracompact) differentiable manifold and this endows

*M*has a complexification, i.e., a complex manifold*X*⊃*M*, dim_{c}*X*= dim_{ℝ}*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*T*ℝ ≅ ℂ given by$${{T}_{\sigma }}\mathbb{R} \mathrel\backepsilon \tau \frac{\partial }{{\partial \sigma }} \leftrightarrow \sigma + i\tau \in \mathbb{C},$$

(1.1)

*T*ℝ 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*T*ℝ depends only on the metric of ℝ. In other words, an isometry of ℝ induces a biholomorphic mapping on*T*ℝ.## 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.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.V. Guillemin and M. Stenzel, Grauert tubes and the homogeneous Monge-Ampère equation,
*J. Diff. Geom.***34**, 561–570 (1991).zbMATHMathSciNetGoogle Scholar - 2.V. Guillemin and M. Stenzel, Grauert tubes and the homogeneous Monge-Ampère equation. II, preprint (1990).Google Scholar
- 3.W. Klingenberg,
*Riemannian Geometry*, Walter de Gruyter, Berlin (1982).zbMATHGoogle Scholar - 4.L. Lempert and R. Szöke, Global solutions on the homogeneous complex Monge-Ampère equation and complex structures on the tangle bundle of Riemannian manifolds,
*Math. Ann.***290**, 689–712 (1991).zbMATHCrossRefMathSciNetGoogle Scholar - 5.G. Patrizio and P. M. Wong, Stein manifolds with compact symmetric centers, preprint (1989).Google Scholar
- 6.B. V. Shabat,
*An Introduction to Complex Analysis*, Nauka, Moscow (1976) (Russian).Google Scholar - 7.R. Szöke, Complex structures on tangent bundles of Riemannian manifolds,
*Math. Ann.***291**, 409–928 (1991).zbMATHCrossRefMathSciNetGoogle Scholar - 8.H. Whitney and F. Bruhat, Quelques propriétés fondamentales des ensembles analytiques-réels,
*Comment. Math. Helv.***33**, 132–160 (1959).zbMATHCrossRefMathSciNetGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 1993