The Complex Structure on Teichmüller Space

Abstract

In this chapter we are going to consider the following question: Does T(M) have a natural almost complex structure? Let us pose this question differently: does A have a natural almost complex structure, and if so, does it pass to the quotient? We will answer these questions in the affirmative. Contrary to the case of two real dimensions, it is not true in higher dimensions that an almost complex structure induces a complex structure. Therefore the next question is whether T(M) has a complex structure. It turns out, it has. In order that an almost complex structure yields a complex structure it is necessary and sufficient that a certain tensor, the Nijenhuis tensor, vanishes identically. The Nijenhuis tensor plays a similar role for this type of question as does the Riemann curvature tensor in Riemannian geometry.KeywordsVector FieldComplex ManifoldRiemannian GeometryCoordinate ChartRiemann Curvature TensorThese 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.