W. M. Goldman 8212 Department of Mathematics, University of Maryland, College Park, Maryland 20742 USA

Zblno.: DE011956362

Author: Po-Hsun Hsieh Title: Linear submanifolds and bisectors in ℂH^{n} Source: Forum Math. 10 (1998), 413–434 Class.: Primary Class.: 53C40 Secondary Class.: Keywords: complex hyperbolic space, totally geodesic submanifold, bisector, linear submanifold, semilinear submanifold Review: This paper (an expanded version of part of the author’s 1997 doctoral thesis at Maryland) concerns (real) submanifolds of complex hyperbolic n-space ℂH^{n} generalizing totally geodesic submanifolds. A submanifold S ⊂ ℂH^{n} is a linear submanifold if it is of the form exp_{o}(V) where o is some point in S and V ⊂ T_{o}ℂH^{n} is a linear subspace. (Necessarily V = T_{o}S.) The point o is called an origin of S. Every point of S is an origin of S if and only if S is totally geodesic. In ℂH^{n} no totally geodesic submanifolds of codimension one exist. Linear submanifolds of codimension one are the (metric) bisectors, defined by the distance function as follows. Choose distinct points x, y ∈ ℂH^{n}. The bisector equidistant from x, y is the subspace S consisting of all z ∈ ℂH^{n} such that d(x, z) = d(y, z). The spine (as defined by Mostow, Pac. J. Math. 86 (1980), 171–276) of S is the intersection of S with the complex geodesic containing x, y. A bisector S is foliated by complex hypersurfaces (dubbed the slices by Mostow) which are orthogonal to the spine. This paper generalizes this theory to linear submanifolds. Consider the maximal complex “subbundle” DL of T L defined as T L ∩ J(T L), where J : T ℂH^{n} ⟶ T ℂH^{n} is the complex structure. If DL has constant dimension and J maps the orthogonal complement of DL ⊂ T L to a subbundle of T ℂH^{n} normal to L, then L is called a CR-submanifold. A linear submanifold is minimal if and only if it is a CR-submanifold. Let L′ be a linear submanifold with an origin o and let L be another linear submanifold with T_{o}L′ ⊂ T_{o}L. Either L ∩ L′ = {o} or L ∩ L′ is a union of geodesics passing through o. Suppose that B_{1}, B_{2} are bisectors whose spines intersect exactly once. Then B_{1} and B_{2} have constant angle along x ∈ B_{1} ∩ B_{2} if and only if B_{1} ∩ B_{2} is a complex submanifold of ℂH^{n}. Intersections of linear submanifolds give rise to semilinear submanifolds, which are defined as the projective image of a ℝ-linear subspace of ℂ^{n,1} for which the restriction of the Hermitian form has index 1. Linear submanifolds are semilinear submanifolds and so are intersections of linear submanifolds. (Hypercycles — equidistant curves — are one-dimensional semilinear submanifolds, called semigeodesics.) Then if a semigeodesic γ intersects a linear submanifold L in at least three points, then γ ⊂ L. The paper generalizes spines and slices of bisectors to linear submanifolds. The spine consists of all A detailed study of 2-dimensional linear submanifolds is given, including formulas for the holomorphy angle of the tangent space T_{p}S and the Gaussing curvature at p in terms of the holomorphy angle of T_{o}S and the distance d(o, p). Remarks: