Almost Complex Surfaces in the Nearly Kähler SL(2,ℝ) × SL(2,ℝ)

The space S L ( 2 , R ) × S L ( 2 , R ) admits a natural homogeneous pseudo-Riemannian nearly Kähler structure. We investigate almost complex surfaces in this space. In particular, we obtain a complete classification of the totally geodesic almost complex surfaces and of the almost complex surfaces with parallel second fundamental form.