Tag Archives: Jacobi field

Index form and Jacobi field

Assume {\gamma(t):[0,a]\rightarrow M} is a normal geodesic, {\gamma(t,s):[0,a]\times[-\epsilon,\epsilon]\rightarrow M} is one parameter smooth variation of curve {\gamma(t)}. Denote {\gamma_s(t)=\gamma(t,s)}.

\displaystyle V(t)=\frac{\partial \gamma_s}{\partial s}\bigg|_{s=0}

is called the variation field. As we all know if the variation is geodesic one, then {V} is a Jacobi field. If {L(s)=L(\gamma_s(t))}, then

\displaystyle L''(0)=\langle\nabla_VV,T\rangle|_0^a+\int_0^a |\dot{V}(t)|^2-R(T,V,T,V)dt

Let us define \mathcal{V}=\{V|V\perp \dot{\gamma}\text{ and } V \text{ is piecewise smooth along }\gamma\} and {\mathcal{V}_0=\{V\in\mathcal{V}|V(0)=V(a)=0\}} is a subspace of {\mathcal{V}}. Index form

\displaystyle I(X,Y)=\int_{0}^a\langle\dot{X},\dot{Y}\rangle-R(T,X,T,Y)dt

for {X,Y\in\mathcal{V}}.

Thm1: If {\gamma} does not have conjugate point, then {I(X,X)>0} for any {X\in \mathcal{V}_0} and {X\neq 0}. This is equivalent to say index form is positive definite.

Thm2(Minimality of Jacobi field): Assume {\gamma} does not have conjugate point. Suppose {V,X\in \mathcal{V}} and {V(0)=X(0)} and {V(a)=X(a)}. If {V} is a Jacobi field on {\gamma}, then

\displaystyle I(V,V)\leq I(X,X)

Thm1 and Thm2 are equivalent.
Thm1 {\Rightarrow} Thm2:

{V-X\in \mathcal{V}}, then

\displaystyle 0\leq I(V-X,V-X)=I(V,V)-2I(V,X)+I(X,X)

\displaystyle =\langle\dot{V},V\rangle|_0^a-2\langle\dot{V},X\rangle|_0^b+I(X,X)=I(V,V)-I(X,X)

Thm2 {\Rightarrow} Thm1: Obviously

One way to prove Thm1 is to prove Thm2. However, we will prove Thm1 directly.

Proof: Suppose {\gamma(t)=\exp_ptv}, {t\in [0,a]}. Since {\gamma} does not have conjugate point, then {d\exp} is non-degenerate in {tv}, {t\in [0,a]}. There exists a neighborhood {U} of {tv} such that {d\exp:U\rightarrow M} is an immersion. From the fact that

{\sigma:[0,a]\rightarrow \exp_pU} is any piecewise smooth curve connecting {\gamma(0)} and {\gamma(a)}, then {L(\gamma)\leq L(\sigma)} when the equality holds, {\sigma} must be monotone reparametirzation of {\gamma}

After proving this, we can construct {\gamma_s\subset \exp_pU} a variation of {\gamma} such that the variation field is {X}, then

\displaystyle 0\leq L''(0)=I(X,X)

Suppose {I(X,X)=0} for some {X\neq 0}, then for any {Y\in \mathcal{V}_0}

\displaystyle 0\leq I(X+\epsilon Y,X+\epsilon Y)

\displaystyle 0\leq I(X-\epsilon Y,X-\epsilon Y)

this means {I(X,Y)=0}, {\forall Y\in \mathcal{V}_0}. This means {X} must be a Jacobi field. But {X\in \mathcal{V}_0} and {\gamma} has no conjugate point, {X} must be 0.

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Meyer’s Thm

Thm:(Meyers) Suppose {M} is a {n-}dimensional complete manifold and

\displaystyle \text{Ric}_M\geq (n-1)K>0

Then {\text{diam}M<\frac{\pi}{\sqrt{K}}} hence {M} is compact and {\pi_1(M)} is finite.

This theorem is proved by the following lemma.

Lemma: Suppose {\text{Ric}_M\geq (n-1)K>0}, then every geodesic which is longer than {\displaystyle \frac{\pi}{\sqrt{K}}} on {M} must have conjugate points hence not a minimal geodesic.

Proof: Suppose {\gamma(t):[0,l]\rightarrow M} is a mininal normal geodesic with length {\displaystyle l>\frac{\pi}{\sqrt{K}}}, {t} is the arc length parameter. Then there exists {\displaystyle K'=\left(\frac{\pi}{l}\right)^2} such that {K'<K}.

For {T_{\gamma(0)}M}, there exists an orthonormal basis {\{\frac{\partial }{\partial t}, V_i, i=1,\cdots,n-1\}}. Let {V_i(t)} be the parallel displacement of {V_i} along {\gamma}, {T} is the one of {\frac{\partial }{\partial t}}.

Define {W_i=V_i(t)\sin \sqrt{K'}t }. Then the second variation of {\gamma} along {W_i} is

\displaystyle I(W_i,W_i)=\int_{0}^l\langle\nabla_TW_i, \nabla_TW_i \rangle+R(T,W_i,T,W_i)dt

\displaystyle =\int_0^l-\langle W_i, \nabla_T\nabla_TW_i\rangle+ R(T,W_i,T,W_i)dt

\displaystyle =\int_0^l K'\sin^2\sqrt{K'}t+\sin^2\sqrt{K'}tR(T,V_i,T,V_i)dt

So

\displaystyle \sum_1^{n-1} I(W_i,W_i)=\int_0^l[(n-1)K'-\text{Ric}_M(V_i)]\sin^2\sqrt{K'}tdt<0

So there exists a {I(W_i,W_i)<0}, for such {i}, this means {\gamma} is locally maximal. This is contradiction, because we choose {\gamma} is a minimal geodesic.