For a Riemannian manifold $(M,g)$, the geodesic flow is $\phi_t:TM\to TM, (x,v)\mapsto (\gamma(t;x,v),\dot{\gamma}(t;x,v))$, where $\gamma(\cdot;x,v)$ is the geodesic started at $x$ with direction $v$. The related vector field of this flow can be formulated as $X(x,v)=(v,0)$.

My question can be viewed as the sham geodesic flow on manifolds without Riemannian metric. Namely, let consider the smooth manifold $TM$ and a vector field

$X:TM\to T(TM),(x,v)\mapsto(v,0)$.

By ODE we know this vector field can integrate to a (local) flow $\phi_t:TM\to TM$ (regardless any Riemannian metric). What is the meaning of this flow? Could we give an explicit formula of this flow?

In the simplese case $M=\mathbb{R}^d$, we know $\phi_t(x,v)=(x+t\cdot v,v)$ (indeed this is the geodesic flow with respect to the flat metric). I do not know if there is some similar form for flows on general manifolds.

Edit: Bill pointed out that the problem is not well formulated. I realized that I misunderstood the canonical splitting into horizontal and vertical parts:

$T(TM)=H\oplus V$ where the horizontal subspace $H$ is the kernel of the connection map $K : T(TM)\to TM$ and the vertical subspace $V = \mathrm{ker}(d\pi)$ is tangent to the fibers of $\pi:TM\to M$.

The vertical subspace $V$ does not involve any metric (hence I got the impression that $V\simeq TM$). At each point $\theta=(x,v)\in TM$, we have $T_\theta(TM)=H_\theta\oplus V_\theta\simeq T_xM\oplus T_xM$. But in general we may not have $V\simeq TM$ or $H\simeq TM$. Is this the point where I went wrong?

For a Riemannian manifold $(M,g)$, the geodesic flow is $\phi_t:TM\to TM, (x,v)\mapsto (\gamma(t;x,v),\dot{\gamma}(t;x,v))$, where $\gamma(\cdot;x,v)$ is the geodesic started at $x$ with direction $v$. The related vector field of this flow can be formulated as $X(x,v)=(v,0)$.

My question can be viewed as the sham geodesic flow on manifolds without Riemannian metric. Namely, let consider the smooth manifold $TM$ and a vector field

$X:TM\to T(TM),(x,v)\mapsto(v,0)$.

By ODE we know this vector field can integrate to a (local) flow $\phi_t:TM\to TM$ (regardless any Riemannian metric). What is the meaning of this flow? Could we give an explicit formula of this flow?

In the simplese case $M=\mathbb{R}^d$, we know $\phi_t(x,v)=(x+t\cdot v,v)$ (indeed this is the geodesic flow with respect to the flat metric). I do not know if there is some similar form for flows on general manifolds.

Edit: Bill pointed out that the problem is not well formulated. I realized that I misunderstood the canonical splitting into horizontal and vertical parts:

$T(TM)=H\oplus V$ where the horizontal subspace $H$ is the kernel of the connection map $K : T(TM)\to TM$ and the vertical subspace $V = \mathrm{ker}(d\pi)$ is tangent to the fibers of $\pi:TM\to M$.

So I think the problem is that the splitting does not make sense if we do not have some metric at hand. And the vector field can not be defined like that.

For a Riemannian manifold $(M,g)$, the geodesic flow is $\phi_t:TM\to TM, (x,v)\mapsto (\gamma(t;x,v),\dot{\gamma}(t;x,v))$, where $\gamma(\cdot;x,v)$ is the geodesic started at $x$ with direction $v$. The related vector field of this flow can be formulated as $X(x,v)=(v,0)$.

My question can be viewed as the sham geodesic flow on manifolds without Riemannian metric. Namely, let consider the smooth manifold $TM$ and a vector field

$X:TM\to T(TM),(x,v)\mapsto(v,0)$.

By ODE we know this vector field can integrate to a (local) flow $\phi_t:TM\to TM$ (regardless any Riemannian metric). What is the meaning of this flow? Could we give an explicit formula of this flow?

In the simplese case $M=\mathbb{R}^d$, we know $\phi_t(x,v)=(x+t\cdot v,v)$ (indeed this is the geodesic flow with respect to the flat metric). I do not know if there is some similar form for flows on general manifolds.

For a Riemannian manifold $(M,g)$, the geodesic flow is $\phi_t:TM\to TM, (x,v)\mapsto (\gamma(t;x,v),\dot{\gamma}(t;x,v))$, where $\gamma(\cdot;x,v)$ is the geodesic started at $x$ with direction $v$. The related vector field of this flow can be formulated as $X(x,v)=(v,0)$.

My question can be viewed as the sham geodesic flow on manifolds without Riemannian metric. Namely, let consider the smooth manifold $TM$ and a vector field

$X:TM\to T(TM),(x,v)\mapsto(v,0)$.

By ODE we know this vector field can integrate to a (local) flow $\phi_t:TM\to TM$ (regardless any Riemannian metric). What is the meaning of this flow? Could we give an explicit formula of this flow?

In the simplese case $M=\mathbb{R}^d$, we know $\phi_t(x,v)=(x+t\cdot v,v)$ (indeed this is the geodesic flow with respect to the flat metric). I do not know if there is some similar form for flows on general manifolds.

Edit: Bill pointed out that the problem is not well formulated. I realized that I misunderstood the canonical splitting into horizontal and vertical parts:

$T(TM)=H\oplus V$ where the horizontal subspace $H$ is the kernel of the connection map $K : T(TM)\to TM$ and the vertical subspace $V = \mathrm{ker}(d\pi)$ is tangent to the fibers of $\pi:TM\to M$.

The vertical subspace $V$ does not involve any metric (hence I got the impression that $V\simeq TM$). At each point $\theta=(x,v)\in TM$, we have $T_\theta(TM)=H_\theta\oplus V_\theta\simeq T_xM\oplus T_xM$. But in general we may not have $V\simeq TM$ or $H\simeq TM$. Is this the point where I went wrong?