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Compactness for Sub-Riemannian Length{minimizers and Subanalyticity

Summary: Compactness for Sub-Riemannian
Length{minimizers and Subanalyticity
Andrei Agrachev 
We establish compactness properties for sets of length{minimizing
admissible paths of a prescribed small length. This implies subanayt-
icity of small sub-Riemannian balls for a wide class of real{analytic
sub-Riemannian structures: for any structure without abnormal mini-
mizers and for many structures without strictly abnormal minimizers.
1 Introduction
Let M be a C 1 Riemannian manifold, dimM = n. A distribution on M is
a smooth linear subbundle  of the tangent bundle TM . We denote by  q
the ber of  at q 2 M ;  q  T q M . A number k = dim q is the rank of the
distribution. We assume that 1 < k < n. The restriction of the Riemannian
structure to  is a sub-Riemannian structure.
Lipschitzian integral curves of the distribution  are called admissible
paths; these are Lipschitzian curves t 7! q(t), t 2 [0; 1], such that _
q(t) 2  q(t)
for almost all t.
We x a point q 0 2 M and study only admissible paths started from this


Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)


Collections: Engineering; Mathematics