Vectors of Higher Rank on a Hadamard Manifold with Compact Quotient

Details Vectors of Higher Rank on a Hadamard Manifold with Compact Quotient Vectors of Higher Rank on a Hadamard Manifold with Compact Quotient

2003-11-03

arxiv.org/abs/math/0311025v1

Mathematics

Differential Geometry

iv + 128 pages, 13 figures, Dissertation Universitaet Zuerich 2003

Scientific paper

Consider a closed, smooth manifold M of nonpositive sectional curvature. Write p:UM-> M for the unit tangent bundle over M and let R_> denote the subset consisting of all vectors of higher rank. This subset is closed and invariant under the geodesic flow on UM. We will define the structured dimension sdim(R_>) which, essentially, is the dimension of the set p(R_>) of base points of R_>. The main result holds for manifolds with sdim(R_>) < dim(M)/2: For every E>0 there is an E-dense, flow invariant, closed subset Z_E in UM R_> such that p(Z_E)=M. For every point in M this means that through this point there is a complete geodesic for which the velocity vector field avoids a neighbourhood of R_>. Gegeben sei eine geschlossene, glatte Mannigfaltigkeit M nichtpositiver Schnittkruemmung. Das Einheitstangentialbuendel sei mit p:UM-> M bezeichnet und die Teilmenge aller Vektoren hoeheren Ranges mit R_>. Diese Teilmenge ist abgeschlossen und invariant unter dem geodaetischen Fluss auf UM. Wir definieren die Strukturdimension sdim(R_>) von R_>, die, im Wesentlichen, die Dimension der Fusspunktmenge p(R_>) misst. Das Hauptergebnis gilt unter der Bedingung, dass sdim(R_>) < dim(M)/2 gilt: Fuer jedes E>0 gibt es eine E-dichte, flussinvariante, abgeschlossene Teilmenge Z_E in UM R_>, fuer die gilt p(Z_E) = M. Dies bedeutet, dass es durch jeden Punkt in M eine vollstaendige Geodaete gibt, deren Geschwindigkeitsfeld eine Umgebung von R_> vermeidet.

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