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Examples of affine maximal torus fibrations of a compact Lie group
by
Marcos Salvai
FaMAF, Universidad Nacional de Córdoba, Argentina
Oral Communication
Abstract. By a generalization of the method developed by Gluck and Warner to characterize the oriented great circle fibrations of the three-sphere, we obtain, for any compact connected semisimple Lie group G, infinite dimensional spaces of examples of smooth and continuous nonsmooth fibrations of G by Weyl-oriented affine maximal tori.
Extended abstract.
Let G be a compact connected semisimple Lie group. A subset S of G is an affine maximal torus if there exist g, h in G such that gSh-1 is a maximal torus of G. Equivalently, it is a maximal connected totally geodesic flat submanifold of G, provided that the group is endowed with a bi-invariant Riemannian metric.
The problem we deal with in this article is, roughly, in which manners (other than the obvious left- or right-invariant ones) G can be expressed as a disjoint union of affine maximal tori. In the spirit of Gluck and Warner, who considered oriented great circle fibrations of the three-sphere, we will study fibrations of G by affine maximal tori which are oriented in the sense of Weyl.
A tangent vector to G is said to be regular if it is tangent to a unique affine maximal torus. An affine Weyl chamber is a connected component of the intersection of TpS with the set of regular tangent vectors, where S is an affine maximal torus and p in S. Given an affine Weyl chamber C, there exists a unique affine maximal torus S such that C is contained in TS. We denote such a torus by \tau(C). The set C of all affine Weyl chambers has a natural fibre bundle structure C --> G.
Recall that a fibration of S3 by oriented great circles is given by a unit vector field on S3 all of whose integral curves are geodesics. Now, a continuous fibration of G by W-oriented affine maximal tori is given by a continuous section \sigma:G --> C such that the (continuous) distribution p --> Tp\tau( \sigma(p) ) is integrable and \tau( \sigma( p)) is a leaf of the distribution (hence the maximal connected leaf) through p for all p in G. If \sigma is smooth, then the set F of W-oriented leaves admits a differentiable structure such that the natural projection G --> F is a smooth fibration.
Fix a maximal torus T. The set T of all W-oriented affine maximal tori of G may be identified in a natural way with ( G/T) ×( G/T).
Consider on G/T any fixed G-invariant Riemannian metric. The first part of the next theorem provides a sufficient condition for a subset of T to be the space of W-oriented fibers of some continuous fibration of G by W-oriented affine maximal tori.
Theorem. For each strictly distance decreasing function f:G/T --> G/T, there exists a unique continuous fibration of G by W-oriented affine maximal tori, such that graph ( f) subset G/T×G/T =~ T is the space of the W-oriented fibers. Moreover, the following assertions are true.
(a) If f is smooth and | df| < 1, then the fibration is smooth.
(b) A partial converse of (a) holds: If the fibration is smooth, then f is smooth.
Remarks.
(a) f= constant =h0T produces the left-invariant fibration with fibers gTh0-1, g in G.
(b) The theorem also holds, with a similar proof, if one substitutes the graph of f with its reflection with respect to the diagonal, i.e. { ( f( y) , y) | y in G/T} . In this case, f= constant =g0T produces the right-invariant fibration with fibers g0Th-1, h in G.
(c) It is an open problem whether a compact connected simple Lie group G =/= S3 admits a fibration such that the set of W-oriented fibers as a subset of T is not the graph of a function from one factor of G/T×G/T to the other. If G is not simple, this is sometimes the case: take for example the group G1×G2 with the fibration whose fibers are ( g, e) ( T1×T2) (e, h-1), where Gj is a compact connected simple Lie group and Tj is a maximal torus of Gj (j=1, 2).
Using the canonical immersion of G/T into the Lie algebra of G, we also
provide, for any G, concrete infinite dimensional spaces of examples of
smooth and continuous nonsmooth fibrations of G as in the theorem. At the
same time, this shows that for any G, the hypothesis
| df| < 1 in item (a) cannot be dropped.
Reference: H. Gluck, F. Warner, Great circle fibrations of
the three-sphere, Duke Math. J. 50 (1983) 107-132.
Date received: May 29, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cadq-81.