# Geometry of Twistor Spaces

## Document Type

Presentation Abstract

## Presentation Date

11-9-2001

## Abstract

The main idea of the twistor theory, created by R. Penrose to solve problems in Mathematical Physics, is that the geometry of a conformal manifold *M* can be "encoded" in holomorphic terms of the so-called twistor space associated to *M*. The Penrose ideas have been developed in the context of the Riemannian geometry by Atiyah, Hitchin and Singer in the case of manifolds of dimension four. In particular, they have defined an almost-complex structure, say *J*_{1}, on the twistor space *Z* of such a manifold *M* which is invariant under conformal changes of the metric of M and found the integrability condition for this almost-complex structure. J. Eells and S. Salamon have introduced another almost-complex structure on *Z*, say *J*_{2}, which, although is not conformally invariant and is never integrable, plays an important role in the harmonic maps theory. The twistor space *Z* admits a natural one-parameter family h_{t}, *t* > 0, of Riemannian metrics compatible with the Atiyah-Hitchin-Singer and Eells- Salamon almost-complex structures. Thus we have two almost-Hermitian manifolds (*Z, J _{1}, h_{t}*) and (

*Z, J*

_{2}, h

_{t}) and the main purpose of this talk is to discuss some geometric properties of these manifolds. More precisely the following topics will be considered:

- The Atiyah-Hitchin-Singer theorem for integrability of the almost-complex structure
*J*_{1}. - The Penrose transform. Applications.
- Twistor spaces with Hermitian Ricci tensor.
- KähIer curvature identies on twistor spaces.

## Recommended Citation

Davidov, Professor Johann, "Geometry of Twistor Spaces" (2001). *Colloquia of the Department of Mathematical Sciences*. 104.

https://scholarworks.umt.edu/mathcolloquia/104

## Additional Details

Friday, 9 November 2001

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)