Mixing
Under exponential map in a Riemannian manifold, what radius keeps the image of a ball a geodesic space?
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Let $M$ be a Riemannian manifold, $pin M$, $T=T_pM$, and $exp_p:Tto M$ the exponential map.
Is there a non-trivial bound from below on the maximum radius of a ball $B_r$ around the origin in $T$, such that $exp_p(B_r)$ is still a geodesic space, that is, for any two points $x,yin T$, a shortest path between $exp_p(x)$ and $exp_p(y)$ lies in $exp_p(B_r)$?
For example, for the sphere of radius $R$, this is $r=frac12pi R$, while the injectivity radius is $pi R$.
Similarly, I expect that the answer would involve the maximum curvature (supposing the curvature on $M$ is bounded from above).
geometry differential-geometry manifolds riemannian-geometry
asked Jan 26 at 19:54
Alexander GelbukhAlexander Gelbukh
1978
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add a comment |
$begingroup$
Let $M$ be a Riemannian manifold, $pin M$, $T=T_pM$, and $exp_p:Tto M$ the exponential map.
Is there a non-trivial bound from below on the maximum radius of a ball $B_r$ around the origin in $T$, such that $exp_p(B_r)$ is still a geodesic space, that is, for any two points $x,yin T$, a shortest path between $exp_p(x)$ and $exp_p(y)$ lies in $exp_p(B_r)$?
For example, for the sphere of radius $R$, this is $r=frac12pi R$, while the injectivity radius is $pi R$.
Similarly, I expect that the answer would involve the maximum curvature (supposing the curvature on $M$ is bounded from above).
geometry differential-geometry manifolds riemannian-geometry
asked Jan 26 at 19:54
Alexander GelbukhAlexander Gelbukh
1978
$endgroup$
$begingroup$
Seems the boundary of such a ball must be a geodesic.
$endgroup$
– Xipan Xiao
Jan 26 at 20:08
$begingroup$
Some googling suggests this is called convexity radius. There seems to be some useful info in this question: math.stackexchange.com/questions/1213366/….
$endgroup$
– Alexander Gelbukh
Jan 26 at 20:12
add a comment |
$begingroup$
Let $M$ be a Riemannian manifold, $pin M$, $T=T_pM$, and $exp_p:Tto M$ the exponential map.
Is there a non-trivial bound from below on the maximum radius of a ball $B_r$ around the origin in $T$, such that $exp_p(B_r)$ is still a geodesic space, that is, for any two points $x,yin T$, a shortest path between $exp_p(x)$ and $exp_p(y)$ lies in $exp_p(B_r)$?
For example, for the sphere of radius $R$, this is $r=frac12pi R$, while the injectivity radius is $pi R$.
Similarly, I expect that the answer would involve the maximum curvature (supposing the curvature on $M$ is bounded from above).
geometry differential-geometry manifolds riemannian-geometry
asked Jan 26 at 19:54
Alexander GelbukhAlexander Gelbukh
1978
$endgroup$
Let $M$ be a Riemannian manifold, $pin M$, $T=T_pM$, and $exp_p:Tto M$ the exponential map.
Is there a non-trivial bound from below on the maximum radius of a ball $B_r$ around the origin in $T$, such that $exp_p(B_r)$ is still a geodesic space, that is, for any two points $x,yin T$, a shortest path between $exp_p(x)$ and $exp_p(y)$ lies in $exp_p(B_r)$?
For example, for the sphere of radius $R$, this is $r=frac12pi R$, while the injectivity radius is $pi R$.
Similarly, I expect that the answer would involve the maximum curvature (supposing the curvature on $M$ is bounded from above).
geometry differential-geometry manifolds riemannian-geometry
geometry differential-geometry manifolds riemannian-geometry
asked Jan 26 at 19:54
Alexander GelbukhAlexander Gelbukh
1978
asked Jan 26 at 19:54
Alexander GelbukhAlexander Gelbukh
1978
asked Jan 26 at 19:54
Alexander GelbukhAlexander Gelbukh
1978
asked Jan 26 at 19:54
Alexander GelbukhAlexander Gelbukh
1978
asked Jan 26 at 19:54
Alexander GelbukhAlexander Gelbukh
1978
1978
$begingroup$
Seems the boundary of such a ball must be a geodesic.
$endgroup$
– Xipan Xiao
Jan 26 at 20:08
$begingroup$
Some googling suggests this is called convexity radius. There seems to be some useful info in this question: math.stackexchange.com/questions/1213366/….
$endgroup$
– Alexander Gelbukh
Jan 26 at 20:12
add a comment |
$begingroup$
Seems the boundary of such a ball must be a geodesic.
$endgroup$
– Xipan Xiao
Jan 26 at 20:08
$begingroup$
Some googling suggests this is called convexity radius. There seems to be some useful info in this question: math.stackexchange.com/questions/1213366/….
$endgroup$
– Alexander Gelbukh
Jan 26 at 20:12
$begingroup$
Seems the boundary of such a ball must be a geodesic.
$endgroup$
– Xipan Xiao
Jan 26 at 20:08
$begingroup$
Seems the boundary of such a ball must be a geodesic.
$endgroup$
– Xipan Xiao
Jan 26 at 20:08
$begingroup$
Some googling suggests this is called convexity radius. There seems to be some useful info in this question: math.stackexchange.com/questions/1213366/….
$endgroup$
– Alexander Gelbukh
Jan 26 at 20:12
$begingroup$
Some googling suggests this is called convexity radius. There seems to be some useful info in this question: math.stackexchange.com/questions/1213366/….
$endgroup$
– Alexander Gelbukh
Jan 26 at 20:12
add a comment |
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$begingroup$
Seems the boundary of such a ball must be a geodesic.
$endgroup$
– Xipan Xiao
Jan 26 at 20:08
$begingroup$
Some googling suggests this is called convexity radius. There seems to be some useful info in this question: math.stackexchange.com/questions/1213366/….
$endgroup$
– Alexander Gelbukh
Jan 26 at 20:12