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

Alexander Nabutovsky; Regina Rotman

Journal of the European Mathematical Society (2003)

- Volume: 005, Issue: 3, page 203-244
- ISSN: 1435-9855

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topNabutovsky, 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 -

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