Intrinsic Directions, Orthogonality, and Distinguished Geodesics in the Symmetrized Bidisc

The symmetrized bidisc $$\begin{aligned} G {\mathop {=}\limits ^\mathrm{{def}}}\{(z+w,zw):|z|<1,\quad |w|<1\}, \end{aligned}$$ G = def { ( z + w , z w ) : | z | < 1 , | w | < 1 } , under the Carathéodory metric, is a complex Finsler space of cohomogeneity 1 in which the geodesics, both real and complex, enjoy a rich geometry. As a Finsler manifold, G does not admit a natural notion of angle, but we nevertheless show that there is a notion of orthogonality. The complex tangent bundle TG splits naturally into the direct sum of two line bundles, which we call the sharp and flat bundles, and which are geometrically defined and therefore covariant under automorphisms of G. Through every point of G, there is a unique complex geodesic of G in the flat direction, having the form $$\begin{aligned} F^\beta {\mathop {=}\limits ^\mathrm{{def}}}\{(\beta +{\bar{\beta }} z,z)\ : z\in \mathbb {D}\} \end{aligned}$$ F β = def { ( β + β ¯ z , z ) : z ∈ D } for some $$\beta \in \mathbb {D}$$ β ∈ D , and called a flat geodesic. We say that a complex geodesic Dis orthogonal to a flat geodesic F if D meets F at a point $$\lambda $$ λ and the complex tangent space $$T_\lambda D$$ T λ D at $$\lambda $$ λ is in the sharp direction at $$\lambda $$ λ . We prove that a geodesic D has the closest point property with respect to a flat geodesic F if and only if D is orthogonal to F in the above sense. Moreover, G is foliated by the geodesics in G that are orthogonal to a fixed flat geodesic F.

A topological obstruction to the removal of a degenerate complex tangent and some related homotopy and homology groups

In this paper, we derive a topological obstruction to the removal of an isolated degenerate complex tangent to an embedding of a 3-manifold into ℂ3 (without affecting the structure of the remaining complex tangents). We demonstrate how the vanishing of this obstruction is both a necessary and sufficient condition for the (local) removal of the isolated complex tangent. The obstruction is a certain homotopy class of the space 𝕐 consisting of totally real 3-planes in the Grassmannian of real 3-planes in ℂ3(= ℝ6). We further compute additional homotopy and homology groups for the space 𝕐 and of its complement 𝕎 consisting of "partially complex" 3-planes in ℂ3.

Complex tangencies to embeddings of Heisenberg groups and odd-dimensional spheres

The notion of a complex tangent arises for embeddings of real manifolds into complex spaces. It is of particular interest when studying embeddings of real n-dimensional manifolds into ℂn. The generic topological structure of the set complex tangents to such embeddings Mn ↪ ℂn takes the form of a (stratified) (n-2)-dimensional submanifold of Mn. In this paper, we generalize our results from our previous work for the 3-dimensional sphere and the Heisenberg group to obtain results regarding the possible topological configurations of the sets of complex tangents to embeddings of odd-dimensional spheres S2n-1 ↪ ℂ2n-1 by first considering the situation for the higher-dimensional analogues of the Heisenberg group.

Weakly complex homogeneous spaces

We complete our recent classification (2011) of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank homogeneous space has weakly complex tangent bundle if and only if it is a product of compact equal rank homogeneous spaces which either carry an invariant almost complex structure (and are classified by Hermann (1955)), or have stably trivial tangent bundle (and are classified by Singhof and Wemmer (1986)), or belong to an explicit list of weakly complex spaces which have neither stably trivial tangent bundle, nor carry invariant almost complex structures.