I am simply going to jump into the proof of the Myers theorem and refer to yesterday’s post for background. In particular, to prove it we only need to prove the following lemma:

Lemma 1 If ${M}$ is a complete Riemannian manifold of dimension ${n}$ and Ricci curvature bounded below by ${C>0}$, then a geodesic of length ${L}$ does not minimize energy if ${L> \pi \sqrt{ \frac{n-1}{C}}}$.

If we choose a geodesic ${\gamma: [0,L] \rightarrow M}$ parametrized by unit length, we then will be done if we find an endpoint-preserving variation ${\gamma_u}$ of ${\gamma}$ with ${\frac{d^2}{d^2 u} E(\gamma_u)_{u=0} > 0}$. In other words if we can find a vector field ${V(t)}$ along ${\gamma}$ with ${V(0)=V(L)=0}$ and

$\displaystyle \boxed{I_2(V) := \int_0^L R\left( \dot{\gamma}, V, \dot{\gamma}, V\right) - g\left( V, \frac{D^2V}{Dt^2} \right) > 0 }$

we will obtain a contradiction; cf. the initial discussion in yesterday’s post and the second variation formula.

Construction of $V$

Choose parallel vector fields ${E_2, \dots, E_n}$ along ${\gamma}$ which, together with ${\gamma'}$, form an orthonormal basis for ${T_{\gamma(t)}(M)}$ at each ${t}$. To do this choose ${E_i(0)}$ for ${i>1}$, and then parallel translate. Then since ${R}$ is skew-symmetric in the last two variables and ${R(\gamma',\gamma',\gamma',\gamma')=0}$

$\displaystyle \rho(\gamma',\gamma') = \sum R( E_i,\gamma', E_i, \gamma' ) \geq C.$

In particular, we could use the sum of the ${E_i}$ to get the vector field ${V}$ satisfying the boxed inequality—except ${E_i(0) \neq 0}$. So we define

$\displaystyle F_i(t) := \sin\left( \frac{\pi t}{L} \right)E_i(t) .$

By the product rule for covariant differentiation and the paralellism of ${E_i}$,

$\displaystyle \frac{D^2}{Dt^2} F_i(t) = -\frac{\pi^2}{L^2} \sin\left( \frac{\pi t}{L} \right)E_i(t).$

So using orthonormality and ${R(F_i, \gamma',F_i,\gamma') \geq C \sin^2\left( \frac{\pi t}{L} \right)}$, we can get that

$\displaystyle \sum I_2(F_i) = \int_0^L C \sin^2\left( \frac{\pi t}{L} \right) - \frac{(n-1)\pi^2}{L^2} \sin^2\left( \frac{\pi t}{L} \right) dt \leq 0$

if ${\gamma}$ minimizes energy, so ${L^2 \leq \frac{(n-1)\pi^2}{C}}$.

Where next?

This ends my MaBloWriMo series on differential geometry; there’s one more day in November, of course, but I need to first learn more new material before I can go further.  Besides, it is probably healthy both for this blog and for myself to cover some other topics.

The posting over the next few weeks will probably be less structured than these entries.  I’m not yet quite sure what I want to discuss, but likely the topic will be one or two out of Riemann surfaces, Koszul complexes and depth, spectral sequences, and singular integrals.  Feel free to suggest something.