I have some vague understanding as following.

First, The isomorphism $\phi_{v}: T_{\theta} TM \longrightarrow T_{p}M \oplus T_{p}M $ is a bridge identifing the splitting of tangent bundle with Jacobi field. Consider curve $\theta(s)=(x(s),v(s))$ in $TM$ with initial vaule $\theta(0)=(x,v)$. So $\frac{d\theta(0)}{ds} \in T_{\theta}TM$. On the other hand,let's denote $J(0):=\frac{dx(0)}{ds}\in T_{p}M$, $D_{t}J(0):=\frac{dv(0)}{ds}\in T_{v}TM \cong T_{p}M$ as the initial value and initial derivatie of a Jacobi field. Now travel $\frac{d\theta(0)}{ds} $ along the differential of the geodesic flow $Dg_{t}$. This yields an isomorphism $ \phi_{g_{t} v}$ taking $Dg_{t}\frac{d\theta(0)}{ds}$ to $(J(t), D_tJ(t))$ which is the value at $t$ of the Jacobi field alone the geodesic corresponding to $g_{t}$ with initial data $J(0), D_{t}J(0)$.

With this identification, it's possible to talk about contracting and expanding of $(J(t),D_tJ(t))$ along $Dg_t$ instead of subbundle of $T_{\theta} TM$. With some effort, one can construct a nice norm $|| \cdot ||$ on $T_{p}M\oplus T_{p}M $ and show $$\frac{\frac{d||J,D_{t}J||}{dt}}{||J,D_{t}J||}>C$$ for some cone of $(J,D_{t}J)$ and some constant $C$.

The key here is to replace $D_{tt}J$ that appears on the upper part of the fraction by curvature which has some restriction is this setting. This suffices to explain expanding and contracting is similar for some other cone.

For more details, please refer to Katok's Introduction to the modern theory of dynamical systems.

I have some vague understanding as following.

First, The isomorphism $\phi_{v}: T_{\theta} TM \longrightarrow T_{p}M \oplus T_{p}M $ is a bridge identifing the splitting of tangent bundle with Jacobi field. Consider curve $\theta(s)=(x(s),v(s))$ in $TM$ with initial vaule $\theta(0)=(x,v)$. So $\frac{d\theta(0)}{ds} \in T_{\theta}TM$. On the other hand,let's denote $J(0):=\frac{dx(0)}{ds}\in T_{p}M$, $D_{t}J(0):=\frac{dv(0)}{ds}\in T_{v}TM \cong T_{p}M$ as the initial value and initial derivatie of a Jacobi field. Now travel $\frac{d\theta(0)}{ds} $ along the differential of the geodesic flow $Dg_{t}$. This yields an isomorphism $ \phi_{g_{t} v}$ taking $Dg_{t}\frac{d\theta(0)}{ds}$ to $(J(t), D_tJ(t))$ which is the value at $t$ of the Jacobi field alone the geodesic corresponding to $g_{t}$ with initial data $J(0), D_{t}J(0)$.

With this identification, it's possible to talk about contracting and expanding of $(J(t),D_tJ(t))$ along $Dg_t$ instead of subbundle of $T_{\theta} TM$. With some effort, one can construct a nice norm $|| \cdot ||$ on $T_{p}M\oplus T_{p}M $ and show $$\frac{\frac{d||J,D_{t}J||}{dt}}{||J,D_{t}J||}>C$$ for some cone of $(J,D_{t}J)$ and some constant $C$.

The key here is to replace $D_{tt}J$ that appears on the upper part of the fraction by curvature which has some restriction in this setting. This suffices to explain expanding. Contracting is similar for some other cone.

For more details, please refer to Katok's Introduction to the modern theory of dynamical systems.

I have some vague understanding as following.

First, The isomorphism $\phi_{v}: T_{\theta} TM \longrightarrow T_{p}M \oplus T_{p}M $ is a bridge identifing the splitting of tangent bundle with Jacobi field. Consider curve $\theta(s)=(x(s),v(s))$ in $TM$ with initial vaule $\theta(0)=(x,v)$. So $\frac{d\theta(0)}{ds} \in T_{\theta}TM$. On the other hand denote $J(0):=\frac{dx(0)}{ds}\in T_{p}M$, $D_{t}J(0):=\frac{dv(0)}{ds}\in T_{v}TM \cong T_{p}M$ as the initial value and initial derivatie of a Jacobi field. Now travel $\frac{d\theta(0)}{ds} $ along the differential of the geodesic flow $Dg_{t}$. This yields an isomorphism $ \phi_{g_{t} v}$ takes $Dg_{t}\frac{d\theta(0)}{ds}$ to $(J(t), D_tJ(t))$ which is the value at $t$ of the Jacobi field alone the geodesic corresponding to $g_{t}$ with initial data $J(0), D_{t}J(0)$.

With this identification, it's possible to talk about contracting and expanding of $(J(t),D_tJ(t))$ along $Dg_t$ instead of subbundle of $T_{\theta} TM$. With some effort, one can construct a nice norm $|| \cdot ||$ on $T_{p}M\oplus T_{p}M $ and show $$\frac{\frac{d||J,D_{t}J||}{dt}}{||J,D_{t}J||}>C$$ for some cone of $(J,D_{t}J)$ and some constant $C$.

The key here is to replace $D_{tt}J$ that appears on the upper part of the fraction by curvature which has some restriction is this setting. This suffices to explain expanding and contracting is similar for some other cone.

For more details, please refer to Katok's Introduction to the modern theory of dynamical systems.

I have some vague understanding as following.

First, The isomorphism $\phi_{v}: T_{\theta} TM \longrightarrow T_{p}M \oplus T_{p}M $ is a bridge identifing the splitting of tangent bundle with Jacobi field. Consider curve $\theta(s)=(x(s),v(s))$ in $TM$ with initial vaule $\theta(0)=(x,v)$. So $\frac{d\theta(0)}{ds} \in T_{\theta}TM$. On the other hand,let's denote $J(0):=\frac{dx(0)}{ds}\in T_{p}M$, $D_{t}J(0):=\frac{dv(0)}{ds}\in T_{v}TM \cong T_{p}M$ as the initial value and initial derivatie of a Jacobi field. Now travel $\frac{d\theta(0)}{ds} $ along the differential of the geodesic flow $Dg_{t}$. This yields an isomorphism $ \phi_{g_{t} v}$ taking $Dg_{t}\frac{d\theta(0)}{ds}$ to $(J(t), D_tJ(t))$ which is the value at $t$ of the Jacobi field alone the geodesic corresponding to $g_{t}$ with initial data $J(0), D_{t}J(0)$.

With this identification, it's possible to talk about contracting and expanding of $(J(t),D_tJ(t))$ along $Dg_t$ instead of subbundle of $T_{\theta} TM$. With some effort, one can construct a nice norm $|| \cdot ||$ on $T_{p}M\oplus T_{p}M $ and show $$\frac{\frac{d||J,D_{t}J||}{dt}}{||J,D_{t}J||}>C$$ for some cone of $(J,D_{t}J)$ and some constant $C$.

The key here is to replace $D_{tt}J$ that appears on the upper part of the fraction by curvature which has some restriction is this setting. This suffices to explain expanding and contracting is similar for some other cone.

For more details, please refer to Katok's Introduction to the modern theory of dynamical systems.