Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem

Nabutovsky, Alexander, and Rotman, Regina. "Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem." Journal of the European Mathematical Society 005.3 (2003): 203-244. <http://eudml.org/doc/277205>.

@article{Nabutovsky2003,
abstract = {In this paper we present two upper bounds on the length of a shortest closed geodesic on compact Riemannian manifolds. The first upper bound depends on an upper bound on sectional curvature and an upper bound on the volume of the manifold. The second upper bound will be given in terms of a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume. The related questions that will also be studied are the following: given a contractible $k$-dimensional sphere in $M^n$, how “fast” can this sphere be contracted to a point, if $\pi _i(M^n)=\lbrace 0\rbrace $ for $1\le i<k$. That is, what is the maximal length of the trajectory described by a point of a sphere under an “optimal” homotopy? Also, what is the “size” of the smallest non-contractible $k$-dimensional sphere in a $(k−1)$-connected manifold $M^n$ providing that $M^n$ is not $k$-connected?},
author = {Nabutovsky, Alexander, Rotman, Regina},
journal = {Journal of the European Mathematical Society},
keywords = {shortest closed geodesic; sectional curvature; volume; smallest non-contractible $k$-dimensional sphere; diameter; shortest closed geodesic; sectional curvature; volume; diameter; smallest non-contractible -dimensional sphere},
language = {eng},
number = {3},
pages = {203-244},
publisher = {European Mathematical Society Publishing House},
title = {Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem},
url = {http://eudml.org/doc/277205},
volume = {005},
year = {2003},
}

TY - JOUR
AU - Nabutovsky, Alexander
AU - Rotman, Regina
TI - Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem
JO - Journal of the European Mathematical Society
PY - 2003
PB - European Mathematical Society Publishing House
VL - 005
IS - 3
SP - 203
EP - 244
AB - In this paper we present two upper bounds on the length of a shortest closed geodesic on compact Riemannian manifolds. The first upper bound depends on an upper bound on sectional curvature and an upper bound on the volume of the manifold. The second upper bound will be given in terms of a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume. The related questions that will also be studied are the following: given a contractible $k$-dimensional sphere in $M^n$, how “fast” can this sphere be contracted to a point, if $\pi _i(M^n)=\lbrace 0\rbrace $ for $1\le i<k$. That is, what is the maximal length of the trajectory described by a point of a sphere under an “optimal” homotopy? Also, what is the “size” of the smallest non-contractible $k$-dimensional sphere in a $(k−1)$-connected manifold $M^n$ providing that $M^n$ is not $k$-connected?
LA - eng
KW - shortest closed geodesic; sectional curvature; volume; smallest non-contractible $k$-dimensional sphere; diameter; shortest closed geodesic; sectional curvature; volume; diameter; smallest non-contractible -dimensional sphere
UR - http://eudml.org/doc/277205
ER -