Lipschitz constant of the exponential map The Next CEO of Stack OverflowDifference between parallel transport and derivative of the exponential mapRelation between exponential map and parallel transportRelation between geodesics and exponential map for Lie groups(locally) “almost convex” property of the distance function in a general Riemannian manifoldLine in product mainifoldCompute distance induced by riemannian metricCovariant derivative of inverse exponential vector fieldHow to use Klingenberg's Lemma to prove Hadamard's theoremKilling field and flow commuting with exponentielLipschitz constant of inner product
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Lipschitz constant of the exponential map
The Next CEO of Stack OverflowDifference between parallel transport and derivative of the exponential mapRelation between exponential map and parallel transportRelation between geodesics and exponential map for Lie groups(locally) “almost convex” property of the distance function in a general Riemannian manifoldLine in product mainifoldCompute distance induced by riemannian metricCovariant derivative of inverse exponential vector fieldHow to use Klingenberg's Lemma to prove Hadamard's theoremKilling field and flow commuting with exponentielLipschitz constant of inner product
$begingroup$
Let $M$ be a smooth Riemannian manifold and let $p in M$. Suppose $r ll textinj(p)$ (the injectivity radius at $p$) and fix $t in (0,r)$ then define the map
$$ T_pM ni v mapsto exp_p(tv) in M$$
I know this map is a diffeomorphism. How can I estimate its Lipschitz constant?
If I take $v_1, v_2 in T_pM$ and I consider the geodesics $gamma_1$ and $gamma_2$ starting at $p$ with initial speed $v_1$ and $v_2$ respectively, it would be enough to prove
$$ fracddt d (gamma_1(t), gamma_2(t)) le |v_2-v_1| $$
to conclude that the Lipschitz constant is bounded by $t$. But even if I consider the differential of the distance given by
$$ fracddt d (gamma_1(t), gamma_2(t))= langle dotalpha_t(d_0), dotgamma_2(t) rangle - langle dotalpha_t(0), dotgamma_1(t) rangle $$
where $d_0 = (gamma_1(t), gamma_2(t))$ and $alpha_t$ is a unit speed geodesic connecting $gamma_1(t)$ to $gamma_2(t)$, I only obtain
beginalign*
fracddt d (gamma_1(t), gamma_2(t)) & = langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)], v_2 rangle - langle mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle \
& = langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)] , v_2 rangle -langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)], v_1 rangle + langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)]-mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle \
& le |v_2-v_1| + langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)]-mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle
endalign*
where $mathcalP^gamma_t_0 to t_1[v]$ is the parallel transport of $v in T_gamma(t_0)$from $gamma(t_0)$ to $gamma(t_1)$ along $gamma$.
It is clear to me that the second summand must go to $0$ as $t to 0^+$ and this would imply the result for small $t$. But I have not idea of how to prove that!
riemannian-geometry lipschitz-functions geodesic
asked yesterday
Bremen000Bremen000
510310
$endgroup$
add a comment |
$begingroup$
Let $M$ be a smooth Riemannian manifold and let $p in M$. Suppose $r ll textinj(p)$ (the injectivity radius at $p$) and fix $t in (0,r)$ then define the map
$$ T_pM ni v mapsto exp_p(tv) in M$$
I know this map is a diffeomorphism. How can I estimate its Lipschitz constant?
If I take $v_1, v_2 in T_pM$ and I consider the geodesics $gamma_1$ and $gamma_2$ starting at $p$ with initial speed $v_1$ and $v_2$ respectively, it would be enough to prove
$$ fracddt d (gamma_1(t), gamma_2(t)) le |v_2-v_1| $$
to conclude that the Lipschitz constant is bounded by $t$. But even if I consider the differential of the distance given by
$$ fracddt d (gamma_1(t), gamma_2(t))= langle dotalpha_t(d_0), dotgamma_2(t) rangle - langle dotalpha_t(0), dotgamma_1(t) rangle $$
where $d_0 = (gamma_1(t), gamma_2(t))$ and $alpha_t$ is a unit speed geodesic connecting $gamma_1(t)$ to $gamma_2(t)$, I only obtain
beginalign*
fracddt d (gamma_1(t), gamma_2(t)) & = langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)], v_2 rangle - langle mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle \
& = langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)] , v_2 rangle -langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)], v_1 rangle + langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)]-mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle \
& le |v_2-v_1| + langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)]-mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle
endalign*
where $mathcalP^gamma_t_0 to t_1[v]$ is the parallel transport of $v in T_gamma(t_0)$from $gamma(t_0)$ to $gamma(t_1)$ along $gamma$.
It is clear to me that the second summand must go to $0$ as $t to 0^+$ and this would imply the result for small $t$. But I have not idea of how to prove that!
riemannian-geometry lipschitz-functions geodesic
asked yesterday
Bremen000Bremen000
510310
$endgroup$
$begingroup$
What is $mathrminj(x)$? Thanks
$endgroup$
– Alex Ortiz
6 hours ago
$begingroup$
It is the infectivity radius at $x$. Basically I am only considering a neighborhood of $0_p in T_pM$ where the exponential map is a diffeomorphism onto some open neighborhood of $p$.
$endgroup$
– Bremen000
6 hours ago
$begingroup$
Yes, as long as you assume an upper sectional curvature bound. This essentially is the content of Rauch Comparison Theorem.
$endgroup$
– Moishe Kohan
2 hours ago
$begingroup$
I have never heard about this result, could you please expand your comment?
$endgroup$
– Bremen000
2 hours ago
add a comment |
$begingroup$
Let $M$ be a smooth Riemannian manifold and let $p in M$. Suppose $r ll textinj(p)$ (the injectivity radius at $p$) and fix $t in (0,r)$ then define the map
$$ T_pM ni v mapsto exp_p(tv) in M$$
I know this map is a diffeomorphism. How can I estimate its Lipschitz constant?
If I take $v_1, v_2 in T_pM$ and I consider the geodesics $gamma_1$ and $gamma_2$ starting at $p$ with initial speed $v_1$ and $v_2$ respectively, it would be enough to prove
$$ fracddt d (gamma_1(t), gamma_2(t)) le |v_2-v_1| $$
to conclude that the Lipschitz constant is bounded by $t$. But even if I consider the differential of the distance given by
$$ fracddt d (gamma_1(t), gamma_2(t))= langle dotalpha_t(d_0), dotgamma_2(t) rangle - langle dotalpha_t(0), dotgamma_1(t) rangle $$
where $d_0 = (gamma_1(t), gamma_2(t))$ and $alpha_t$ is a unit speed geodesic connecting $gamma_1(t)$ to $gamma_2(t)$, I only obtain
beginalign*
fracddt d (gamma_1(t), gamma_2(t)) & = langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)], v_2 rangle - langle mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle \
& = langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)] , v_2 rangle -langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)], v_1 rangle + langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)]-mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle \
& le |v_2-v_1| + langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)]-mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle
endalign*
where $mathcalP^gamma_t_0 to t_1[v]$ is the parallel transport of $v in T_gamma(t_0)$from $gamma(t_0)$ to $gamma(t_1)$ along $gamma$.
It is clear to me that the second summand must go to $0$ as $t to 0^+$ and this would imply the result for small $t$. But I have not idea of how to prove that!
riemannian-geometry lipschitz-functions geodesic
asked yesterday
Bremen000Bremen000
510310
$endgroup$
Let $M$ be a smooth Riemannian manifold and let $p in M$. Suppose $r ll textinj(p)$ (the injectivity radius at $p$) and fix $t in (0,r)$ then define the map
$$ T_pM ni v mapsto exp_p(tv) in M$$
I know this map is a diffeomorphism. How can I estimate its Lipschitz constant?
If I take $v_1, v_2 in T_pM$ and I consider the geodesics $gamma_1$ and $gamma_2$ starting at $p$ with initial speed $v_1$ and $v_2$ respectively, it would be enough to prove
$$ fracddt d (gamma_1(t), gamma_2(t)) le |v_2-v_1| $$
to conclude that the Lipschitz constant is bounded by $t$. But even if I consider the differential of the distance given by
$$ fracddt d (gamma_1(t), gamma_2(t))= langle dotalpha_t(d_0), dotgamma_2(t) rangle - langle dotalpha_t(0), dotgamma_1(t) rangle $$
where $d_0 = (gamma_1(t), gamma_2(t))$ and $alpha_t$ is a unit speed geodesic connecting $gamma_1(t)$ to $gamma_2(t)$, I only obtain
beginalign*
fracddt d (gamma_1(t), gamma_2(t)) & = langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)], v_2 rangle - langle mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle \
& = langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)] , v_2 rangle -langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)], v_1 rangle + langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)]-mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle \
& le |v_2-v_1| + langle mathcalP^gamma_2_t to 0[dotalpha_t(d_0)]-mathcalP^gamma_1_t to 0[dotalpha_t(0)], v_1 rangle
endalign*
where $mathcalP^gamma_t_0 to t_1[v]$ is the parallel transport of $v in T_gamma(t_0)$from $gamma(t_0)$ to $gamma(t_1)$ along $gamma$.
It is clear to me that the second summand must go to $0$ as $t to 0^+$ and this would imply the result for small $t$. But I have not idea of how to prove that!
riemannian-geometry lipschitz-functions geodesic
riemannian-geometry lipschitz-functions geodesic
asked yesterday
Bremen000Bremen000
510310
asked yesterday
Bremen000Bremen000
510310
edited 2 hours ago
Bremen000
asked yesterday
Bremen000Bremen000
510310
asked yesterday
Bremen000Bremen000
510310
asked yesterday
Bremen000Bremen000
510310
510310
$begingroup$
What is $mathrminj(x)$? Thanks
$endgroup$
– Alex Ortiz
6 hours ago
$begingroup$
It is the infectivity radius at $x$. Basically I am only considering a neighborhood of $0_p in T_pM$ where the exponential map is a diffeomorphism onto some open neighborhood of $p$.
$endgroup$
– Bremen000
6 hours ago
$begingroup$
Yes, as long as you assume an upper sectional curvature bound. This essentially is the content of Rauch Comparison Theorem.
$endgroup$
– Moishe Kohan
2 hours ago
$begingroup$
I have never heard about this result, could you please expand your comment?
$endgroup$
– Bremen000
2 hours ago
add a comment |
$begingroup$
What is $mathrminj(x)$? Thanks
$endgroup$
– Alex Ortiz
6 hours ago
$begingroup$
It is the infectivity radius at $x$. Basically I am only considering a neighborhood of $0_p in T_pM$ where the exponential map is a diffeomorphism onto some open neighborhood of $p$.
$endgroup$
– Bremen000
6 hours ago
$begingroup$
Yes, as long as you assume an upper sectional curvature bound. This essentially is the content of Rauch Comparison Theorem.
$endgroup$
– Moishe Kohan
2 hours ago
$begingroup$
I have never heard about this result, could you please expand your comment?
$endgroup$
– Bremen000
2 hours ago
$begingroup$
What is $mathrminj(x)$? Thanks
$endgroup$
– Alex Ortiz
6 hours ago
$begingroup$
What is $mathrminj(x)$? Thanks
$endgroup$
– Alex Ortiz
6 hours ago
$begingroup$
It is the infectivity radius at $x$. Basically I am only considering a neighborhood of $0_p in T_pM$ where the exponential map is a diffeomorphism onto some open neighborhood of $p$.
$endgroup$
– Bremen000
6 hours ago
$begingroup$
It is the infectivity radius at $x$. Basically I am only considering a neighborhood of $0_p in T_pM$ where the exponential map is a diffeomorphism onto some open neighborhood of $p$.
$endgroup$
– Bremen000
6 hours ago
$begingroup$
Yes, as long as you assume an upper sectional curvature bound. This essentially is the content of Rauch Comparison Theorem.
$endgroup$
– Moishe Kohan
2 hours ago
$begingroup$
Yes, as long as you assume an upper sectional curvature bound. This essentially is the content of Rauch Comparison Theorem.
$endgroup$
– Moishe Kohan
2 hours ago
$begingroup$
I have never heard about this result, could you please expand your comment?
$endgroup$
– Bremen000
2 hours ago
$begingroup$
I have never heard about this result, could you please expand your comment?
$endgroup$
– Bremen000
2 hours ago
add a comment |
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$begingroup$
What is $mathrminj(x)$? Thanks
$endgroup$
– Alex Ortiz
6 hours ago
$begingroup$
It is the infectivity radius at $x$. Basically I am only considering a neighborhood of $0_p in T_pM$ where the exponential map is a diffeomorphism onto some open neighborhood of $p$.
$endgroup$
– Bremen000
6 hours ago
$begingroup$
Yes, as long as you assume an upper sectional curvature bound. This essentially is the content of Rauch Comparison Theorem.
$endgroup$
– Moishe Kohan
2 hours ago
$begingroup$
I have never heard about this result, could you please expand your comment?
$endgroup$
– Bremen000
2 hours ago