Geodesically equivalent metrics on homogenous spaces

Bokan, Neda, Šukilović, Tijana, and Vukmirović, Srdjan. "Geodesically equivalent metrics on homogenous spaces." Czechoslovak Mathematical Journal 69.4 (2019): 945-954. <http://eudml.org/doc/294279>.

@article{Bokan2019,
abstract = {Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two $G$-invariant metrics of arbitrary signature on homogenous space $G/H$ are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that the existence of nonproportional, geodesically equivalent, $G$-invariant metrics on homogenous space $G/H$ implies that their holonomy algebra cannot be full. We give an algorithm for finding all left invariant metrics geodesically equivalent to a given left invariant metric on a Lie group. Using that algorithm we prove that no two left invariant metrics of any signature on sphere $S^3$ are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, nonproportional, left-invariant metrics.},
author = {Bokan, Neda, Šukilović, Tijana, Vukmirović, Srdjan},
journal = {Czechoslovak Mathematical Journal},
keywords = {invariant metric; geodesically equivalent metric; affinely equivalent metric},
language = {eng},
number = {4},
pages = {945-954},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Geodesically equivalent metrics on homogenous spaces},
url = {http://eudml.org/doc/294279},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Bokan, Neda
AU - Šukilović, Tijana
AU - Vukmirović, Srdjan
TI - Geodesically equivalent metrics on homogenous spaces
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 4
SP - 945
EP - 954
AB - Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two $G$-invariant metrics of arbitrary signature on homogenous space $G/H$ are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that the existence of nonproportional, geodesically equivalent, $G$-invariant metrics on homogenous space $G/H$ implies that their holonomy algebra cannot be full. We give an algorithm for finding all left invariant metrics geodesically equivalent to a given left invariant metric on a Lie group. Using that algorithm we prove that no two left invariant metrics of any signature on sphere $S^3$ are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, nonproportional, left-invariant metrics.
LA - eng
KW - invariant metric; geodesically equivalent metric; affinely equivalent metric
UR - http://eudml.org/doc/294279
ER -