Geodesics of Sasaki metric

Solution 1:

(1) Consider a projection $\pi : (T_1M,G) \rightarrow (M,g) $ We want to define a metric $G$. If $\widetilde{c_1} (t),\ \widetilde{c_2}(t) $ are curves starting at $(p,v)$ then we write $\widetilde{c_i}(t)=(c_i(t),v_i(t)),\ c_i(0)=p,\ v_i(0)=v$ so that let $$ G(\widetilde{c_1}'(0),\widetilde{c_2}'(0)) =g(c_1'(0),c_2'(0)) + g(\nabla_{c_1'} v_1,\nabla_{c_2'} v_2)\ \ast $$

And if $c_1(t)=p$, then we use $\frac{d}{dt}\bigg|_{t=0}\ v_1(t)$ instead of $\nabla_{c_1'} v_1$.

(2) Now we will interpret this definition. Fix a curve $c$ at $c(0)=p\in M$. If $v(t)$ is a parallel vector field along $c(t)$ with $v(0)=v$, then define $\widetilde{c}(t)=(c(t),v(t))$ So we have ${\rm length}\ c ={\rm length}\ \widetilde{c}$. That is any curve in $M$ can be lifted into some curve of same length to any point.

(3) If $c$ is a unit speed geodesic in $M$, then $\widetilde{c}(t)=(c(t),c'(t))$, lift of $c$, is a geodesic in $T_1M$ : Let $\widetilde{p}=\widetilde{c}(0),\ \widetilde{q}=\widetilde{c}(\epsilon )$ Assume that $\widetilde{\gamma} =(\gamma (t),v(t))$ is a geodesic in $T_1M$ between two points s.t. $\gamma$ has unit speed. Hence since $${\rm length}\ \gamma\geq {\rm length}\ c,$$ then $\widetilde{\gamma}$ is $\widetilde{c}$