Given a Riemannian manifold $X$, a point $x\in X$ and $u,v\in T_xX$, I wanted to compare the following three vectors in $T_{\exp_x(v)}X$.

$u_1=$ The parallel transport of $u$ along the geodesic $\exp_x(tv)$ at $t=1$;

$u_2=(d\exp_x)|_{\exp_x(v)}u$, the push-forward of $u$ under the exponential map $T_xX\rightarrow X$ at $\exp_x(v)$;

$u_3$: Consider the vector field $v_\tau$ along the geodesic $\exp_x(\tau u)$ by parallel transport of $v$, then take the geodesic $\gamma_\tau(t)$ determined by $v_\tau$, thus we have a family of geodesics. Now take the Jacobi vector field at $t=1$ to be $u_3$, i.e. $\frac{d}{d\tau}|_{\tau=0}(\gamma_\tau(1))$.

My question is

Does $\sup\limits_{\|u\|=1}\|u_i-u_j\|^2\leq o(\|v\|^2)$ hold near $x$?

If it is not true in general, what conditions should one put on the metric to ensure this?

Thanks in advance!

  • $\begingroup$ Can you clarify the definition of $u_3$? What's $w$ and how is $u$ involved? $\endgroup$ – Anthony Carapetis Oct 19 '14 at 6:55
  • $\begingroup$ @AnthonyCarapetis, sorry that was a typo, $w$ meant $u$. I just corrected it. $\endgroup$ – Xin Jin Oct 19 '14 at 15:52
  • $\begingroup$ The answer is true. One can use geodesic coordinate to deduce that. $\endgroup$ – Xin Jin Oct 20 '14 at 2:55

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