On Eisenhart’s Type Theorem for Sub-Riemannian Metrics on Step $2$ Distributions with $\mathrm{ad}$-Surjective Tanaka Symbols
Zaifeng Lin,
Igor Zelenko Department of Mathematics, Texas A\&M University,
TX 77843 College Station, USA
Аннотация:
The classical result of Eisenhart states that, if a Riemannian metric
$g$ admits a Riemannian metric that is not constantly proportional to
$g$ and has the same (parameterized) geodesics as
$g$ in a neighborhood of a given point, then
$g$ is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step
$2$ graded nilpotent Lie algebras, called \emph{
$\mathrm{ad}$-surjective}, and extend the Eisenhart theorem to sub-Riemannian metrics on step
$2$ distributions with
$\mathrm{ad}$-surjective Tanaka symbols. The class of
ad-surjective step
$2$ nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case.
Ключевые слова:
sub-Riemannian geometry, Riemannian geometry, sub-Riemannian Geodesics, separation of variables, nilpotent approximation, Tanaka symbol, orbital equivalence, overdetermined PDEs, graded nilpotent Lie algebras
MSC: 53C17,
58A30,
58E10,
53A15,
37J39,
35N10,
17B70 Поступила в редакцию: 05.09.2023
Принята в печать: 04.01.2024
Язык публикации: английский
DOI:
10.1134/S1560354724020023