Proof of the Symmetry Lemma

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I'm hoping someone can check that my proof of the Symmetry Lemma below is okay. (I'm actually proving a version of the result restricted to geodesic variations, because that's what I'm primarily interested in.) Note: I have seen the proof done with local coordinates, e.g., in John M. Lee's book; I want to avoid using coordinates.

Let $gamma : [0,T] to M$ be a geodesic in a Riemannian manifold $(M,g)$, and let $sigma : (-varepsilon,varepsilon) times [0,T] to M$, $(s,t) mapsto sigma(s,t)$ be a smooth geodesic variation of $gamma$ (i.e., $sigma(0,cdot) = gamma$, and $sigma(s,cdot)$ is a geodesic for every $s$). We can define vector fields along $sigma$ as follows:
$$
J(s,t) = T_(s,t)sigmacdotleft.fracpartialpartial sright|_(s,t)
=: fracpartialsigmapartial s(s,t)
quadtextandquad
Z(s,t) = T_(s,t)sigmacdotleft.fracpartialpartial tright|_(s,t)
=: fracpartialsigmapartial t(s,t).
$$
(Here $s$ is the coordinate on $(-varepsilon,varepsilon)$, and $t$ is that on $[0,T]$.) So we have $dotgamma = Z(0,cdot)$, and $J(0,cdot)$ is a Jacobi field along $gamma$.

Claim: $fracDdsZ = fracDdtJ$.

Proof: Let $widetildeJ,widetildeZ in mathfrakX(M)$ be (smooth) extensions of $J$ and $Z$, resp., to an open neighbourhood of $sigma$, i.e., $widetildeJ(sigma(s,t)) = J(s,t)$ and $widetildeZ(sigma(s,t)) = Z(s,t)$. Since $partial/partial s$ and $widetildeJ$ are $sigma$-related, likewise $partial/partial t$ and $widetildeZ$, we then get that
$$
[widetildeJ,widetildeZ]circsigma
= Tsigmacdotleft[fracpartialpartial s,fracpartialpartial tright]
= 0.
$$
Thus, since the Levi-Civita connection $nabla$ is symmetric:
beginalign*
0 & = T(J(s,t),Z(s,t))\
& = T(widetildeJ,widetildeZ)(sigma(s,t))\
& = nabla_widetildeJwidetildeZ(sigma(s,t)) - nabla_widetildeZwidetildeJ(sigma(s,t)) - [widetildeJ,widetildeZ](sigma(s,t))\
& = nabla_J(s,t)(widetildeZcircsigma) - nabla_Z(s,t)(widetildeJcircsigma)\
& = fracDdsZ(s,t) - fracDdtJ(s,t).
endalign*
(End of proof.)

My main concern with this argument is the existence of the extensions $widetildeJ$ and $widetildeZ$ (and hence that $[widetildeJ,widetildeZ]circsigma = 0$). Clearly they won't exist on the entire image of $sigma$, but if we restrict to a sufficiently small open subset $A subset (-varepsilon,varepsilon) times [0,T]$, then can we be assured that they exist on $sigma(A)$? (Everything else in the proof seems okay to me, but I could be mistaken.)

Edit: my claim was originally written incorrectly ($J$ and $Z$ were on the wrong sides); I've corrected this.

differential-geometry riemannian-geometry

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edited Jul 25 at 9:20

asked Jul 24 at 11:02

Den

443

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up vote
6
down vote

favorite

3

I'm hoping someone can check that my proof of the Symmetry Lemma below is okay. (I'm actually proving a version of the result restricted to geodesic variations, because that's what I'm primarily interested in.) Note: I have seen the proof done with local coordinates, e.g., in John M. Lee's book; I want to avoid using coordinates.

Let $gamma : [0,T] to M$ be a geodesic in a Riemannian manifold $(M,g)$, and let $sigma : (-varepsilon,varepsilon) times [0,T] to M$, $(s,t) mapsto sigma(s,t)$ be a smooth geodesic variation of $gamma$ (i.e., $sigma(0,cdot) = gamma$, and $sigma(s,cdot)$ is a geodesic for every $s$). We can define vector fields along $sigma$ as follows:
$$
J(s,t) = T_(s,t)sigmacdotleft.fracpartialpartial sright|_(s,t)
=: fracpartialsigmapartial s(s,t)
quadtextandquad
Z(s,t) = T_(s,t)sigmacdotleft.fracpartialpartial tright|_(s,t)
=: fracpartialsigmapartial t(s,t).
$$
(Here $s$ is the coordinate on $(-varepsilon,varepsilon)$, and $t$ is that on $[0,T]$.) So we have $dotgamma = Z(0,cdot)$, and $J(0,cdot)$ is a Jacobi field along $gamma$.

Claim: $fracDdsZ = fracDdtJ$.

Proof: Let $widetildeJ,widetildeZ in mathfrakX(M)$ be (smooth) extensions of $J$ and $Z$, resp., to an open neighbourhood of $sigma$, i.e., $widetildeJ(sigma(s,t)) = J(s,t)$ and $widetildeZ(sigma(s,t)) = Z(s,t)$. Since $partial/partial s$ and $widetildeJ$ are $sigma$-related, likewise $partial/partial t$ and $widetildeZ$, we then get that
$$
[widetildeJ,widetildeZ]circsigma
= Tsigmacdotleft[fracpartialpartial s,fracpartialpartial tright]
= 0.
$$
Thus, since the Levi-Civita connection $nabla$ is symmetric:
beginalign*
0 & = T(J(s,t),Z(s,t))\
& = T(widetildeJ,widetildeZ)(sigma(s,t))\
& = nabla_widetildeJwidetildeZ(sigma(s,t)) - nabla_widetildeZwidetildeJ(sigma(s,t)) - [widetildeJ,widetildeZ](sigma(s,t))\
& = nabla_J(s,t)(widetildeZcircsigma) - nabla_Z(s,t)(widetildeJcircsigma)\
& = fracDdsZ(s,t) - fracDdtJ(s,t).
endalign*
(End of proof.)

My main concern with this argument is the existence of the extensions $widetildeJ$ and $widetildeZ$ (and hence that $[widetildeJ,widetildeZ]circsigma = 0$). Clearly they won't exist on the entire image of $sigma$, but if we restrict to a sufficiently small open subset $A subset (-varepsilon,varepsilon) times [0,T]$, then can we be assured that they exist on $sigma(A)$? (Everything else in the proof seems okay to me, but I could be mistaken.)

Edit: my claim was originally written incorrectly ($J$ and $Z$ were on the wrong sides); I've corrected this.

differential-geometry riemannian-geometry

share|cite|improve this question

edited Jul 25 at 9:20

asked Jul 24 at 11:02

Den

443

add a comment | 

up vote
6
down vote

favorite

3

up vote
6
down vote

favorite

3

3

I'm hoping someone can check that my proof of the Symmetry Lemma below is okay. (I'm actually proving a version of the result restricted to geodesic variations, because that's what I'm primarily interested in.) Note: I have seen the proof done with local coordinates, e.g., in John M. Lee's book; I want to avoid using coordinates.

Let $gamma : [0,T] to M$ be a geodesic in a Riemannian manifold $(M,g)$, and let $sigma : (-varepsilon,varepsilon) times [0,T] to M$, $(s,t) mapsto sigma(s,t)$ be a smooth geodesic variation of $gamma$ (i.e., $sigma(0,cdot) = gamma$, and $sigma(s,cdot)$ is a geodesic for every $s$). We can define vector fields along $sigma$ as follows:
$$
J(s,t) = T_(s,t)sigmacdotleft.fracpartialpartial sright|_(s,t)
=: fracpartialsigmapartial s(s,t)
quadtextandquad
Z(s,t) = T_(s,t)sigmacdotleft.fracpartialpartial tright|_(s,t)
=: fracpartialsigmapartial t(s,t).
$$
(Here $s$ is the coordinate on $(-varepsilon,varepsilon)$, and $t$ is that on $[0,T]$.) So we have $dotgamma = Z(0,cdot)$, and $J(0,cdot)$ is a Jacobi field along $gamma$.

Claim: $fracDdsZ = fracDdtJ$.

Proof: Let $widetildeJ,widetildeZ in mathfrakX(M)$ be (smooth) extensions of $J$ and $Z$, resp., to an open neighbourhood of $sigma$, i.e., $widetildeJ(sigma(s,t)) = J(s,t)$ and $widetildeZ(sigma(s,t)) = Z(s,t)$. Since $partial/partial s$ and $widetildeJ$ are $sigma$-related, likewise $partial/partial t$ and $widetildeZ$, we then get that
$$
[widetildeJ,widetildeZ]circsigma
= Tsigmacdotleft[fracpartialpartial s,fracpartialpartial tright]
= 0.
$$
Thus, since the Levi-Civita connection $nabla$ is symmetric:
beginalign*
0 & = T(J(s,t),Z(s,t))\
& = T(widetildeJ,widetildeZ)(sigma(s,t))\
& = nabla_widetildeJwidetildeZ(sigma(s,t)) - nabla_widetildeZwidetildeJ(sigma(s,t)) - [widetildeJ,widetildeZ](sigma(s,t))\
& = nabla_J(s,t)(widetildeZcircsigma) - nabla_Z(s,t)(widetildeJcircsigma)\
& = fracDdsZ(s,t) - fracDdtJ(s,t).
endalign*
(End of proof.)

My main concern with this argument is the existence of the extensions $widetildeJ$ and $widetildeZ$ (and hence that $[widetildeJ,widetildeZ]circsigma = 0$). Clearly they won't exist on the entire image of $sigma$, but if we restrict to a sufficiently small open subset $A subset (-varepsilon,varepsilon) times [0,T]$, then can we be assured that they exist on $sigma(A)$? (Everything else in the proof seems okay to me, but I could be mistaken.)

Edit: my claim was originally written incorrectly ($J$ and $Z$ were on the wrong sides); I've corrected this.

differential-geometry riemannian-geometry

share|cite|improve this question

edited Jul 25 at 9:20

asked Jul 24 at 11:02

Den

443

I'm hoping someone can check that my proof of the Symmetry Lemma below is okay. (I'm actually proving a version of the result restricted to geodesic variations, because that's what I'm primarily interested in.) Note: I have seen the proof done with local coordinates, e.g., in John M. Lee's book; I want to avoid using coordinates.

Let $gamma : [0,T] to M$ be a geodesic in a Riemannian manifold $(M,g)$, and let $sigma : (-varepsilon,varepsilon) times [0,T] to M$, $(s,t) mapsto sigma(s,t)$ be a smooth geodesic variation of $gamma$ (i.e., $sigma(0,cdot) = gamma$, and $sigma(s,cdot)$ is a geodesic for every $s$). We can define vector fields along $sigma$ as follows:
$$
J(s,t) = T_(s,t)sigmacdotleft.fracpartialpartial sright|_(s,t)
=: fracpartialsigmapartial s(s,t)
quadtextandquad
Z(s,t) = T_(s,t)sigmacdotleft.fracpartialpartial tright|_(s,t)
=: fracpartialsigmapartial t(s,t).
$$
(Here $s$ is the coordinate on $(-varepsilon,varepsilon)$, and $t$ is that on $[0,T]$.) So we have $dotgamma = Z(0,cdot)$, and $J(0,cdot)$ is a Jacobi field along $gamma$.

Claim: $fracDdsZ = fracDdtJ$.

Proof: Let $widetildeJ,widetildeZ in mathfrakX(M)$ be (smooth) extensions of $J$ and $Z$, resp., to an open neighbourhood of $sigma$, i.e., $widetildeJ(sigma(s,t)) = J(s,t)$ and $widetildeZ(sigma(s,t)) = Z(s,t)$. Since $partial/partial s$ and $widetildeJ$ are $sigma$-related, likewise $partial/partial t$ and $widetildeZ$, we then get that
$$
[widetildeJ,widetildeZ]circsigma
= Tsigmacdotleft[fracpartialpartial s,fracpartialpartial tright]
= 0.
$$
Thus, since the Levi-Civita connection $nabla$ is symmetric:
beginalign*
0 & = T(J(s,t),Z(s,t))\
& = T(widetildeJ,widetildeZ)(sigma(s,t))\
& = nabla_widetildeJwidetildeZ(sigma(s,t)) - nabla_widetildeZwidetildeJ(sigma(s,t)) - [widetildeJ,widetildeZ](sigma(s,t))\
& = nabla_J(s,t)(widetildeZcircsigma) - nabla_Z(s,t)(widetildeJcircsigma)\
& = fracDdsZ(s,t) - fracDdtJ(s,t).
endalign*
(End of proof.)

My main concern with this argument is the existence of the extensions $widetildeJ$ and $widetildeZ$ (and hence that $[widetildeJ,widetildeZ]circsigma = 0$). Clearly they won't exist on the entire image of $sigma$, but if we restrict to a sufficiently small open subset $A subset (-varepsilon,varepsilon) times [0,T]$, then can we be assured that they exist on $sigma(A)$? (Everything else in the proof seems okay to me, but I could be mistaken.)

Edit: my claim was originally written incorrectly ($J$ and $Z$ were on the wrong sides); I've corrected this.

differential-geometry riemannian-geometry

share|cite|improve this question

edited Jul 25 at 9:20

asked Jul 24 at 11:02

Den

443

share|cite|improve this question

share|cite|improve this question

edited Jul 25 at 9:20

edited Jul 25 at 9:20

edited Jul 25 at 9:20

asked Jul 24 at 11:02

Den

443

asked Jul 24 at 11:02

Den

443

asked Jul 24 at 11:02

Den

443

443

add a comment | 

add a comment | 

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