zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Sub-Riemannian geometry. (English) Zbl 0609.53021

A sub-Riemannian or singular Riemannian geometry is given by a smoothly varying positive definite quadratic form defined only on a subbundle S of the tangent bundle TM of a differentiable manifold, S being bracket-generating, that is sections of S together with their Lie brackets generate the C (M)-module V(M) of vector fields. Such a metric is also called aCarnot-Carathéodory metric.

The paper gives a fairly self-contained description of the known results concerning length minimizing curves and geodesics, the exponential map, completeness and isometries, the latter allowing the investigation of sub-Riemannian symmetric spaces, which, in the case of three dimensions, are completely classified up to local isometry.

For piecewise smooth curves tangent to the subbundle S there is a natural notion of length. If M is connected, by a theorem of Chow (1939) it is possible to connect any two points of M by such a curve, giving M the structure of a metric space. The sub-Riemannian metric gives a natural positive semidefinite quadratic form on covectors and therefore a quadratic function on the cotangent bundle, the Hamiltonian vector field of which replaces the geodesic spray of a Riemannian manifold. The projections of the integrals of this vectorfield are the geodesics of the sub-Riemannian structure. The main results concerning geodesics are: Every length-minimizing curve is a geodesic and locally every geodesic is length-minimizing, the latter result as some others only being derived under the so-called strong bracket generating hypothesis, that is the sections of S and the brackets of such sections with any non-zero section of S generate V(M).

The proof requires a careful study of the exponential map which can be defined in a natural way as a map from T * M to M. In contrast to the Riemannian situation the exponential map never is a diffeomorphism at the origin. Another basic result is the analogue of the Hopf-Rinow theorem, the completeness of M being proved under the mentioned hypothesis. Possibly for some results this hypothesis can be dropped, but, as the exponential map looses some of its properties, other methods will be required.

Corollary 6.2 and the other results depending on it have to be modified because the original proof is not correct. Also some other corrections are indicated.

Reviewer: K.Horneffer

53C99Global differential geometry
93B99Controllability, observability, and system structure