Riemannian metrics on the tangent bundle For an application I'm working on, I'm interested in defining a metric on the tangle bundle $TM$ of a Riemannian manifold $(M, g)$. Let $g$ be given in local coordinates as $d\sigma^2_M = g_{ij} dx^i dx^j$. I was wondering if there is anything wrong with defining a metric on the tangent bundle as $d\sigma^2_{TM} = g_{ij}dx^idx^j + g_{ij} dv^i dv^j$ where $g_{ij}$ comes directly from $g$ on $M$ itself. I am aware of the definition of the Sasaki metric and that it is a more "natural" metric. For my application, defining $(TM, \tilde{g})$ in this way makes things simple, but I have not seen it anywhere and I am wondering if there is a reason why.
 A: In short, your metric is coordinate dependent; different charts will in general produce a different metric on their common domain. It takes a bit of computation to show this.
Let $x^i$ and $\bar{x}^i$ be coordinates for two overlapping charts (with the indexless $x$, $\bar{x}$ denoting the entire tuple) with a transition function $\varphi(\bar{x})=x$, and denote the Jacobian of this function $\Phi^i_j=\frac{\partial x^i}{\partial \bar{x}^j}$ We can write the metric in these coordinates as $g$ and $\bar{g}$ respectively, related by $\bar{g}=\varphi^* g$, or equivalently $\bar{g}_{ij}=\Phi^k_i\Phi^l_j g_{kl}$.
These coordintes induce coordinates $(x,v)$ and $(\bar{x},\bar{v})$ on $TM$ in the standard way. Above a particular point, we have $\bar{v}^j\bar{\partial}_j=\bar{v}^j\Phi^i_j\partial_j$, and thus $v^i=\Phi^i_j\bar{v}^j$. Using this, we can write out the transition function and all of its derivatives:
$$\begin{align}
x^i&=\varphi^i(\bar{x}) &\ \ \ \ \ \ \ \ 
v^i&=\Phi^i_j(\bar{x})\bar{v}^j \\
\frac{\partial x^i}{\partial\bar{x}^j}&=\Phi^i_j(\bar{x}) &\ \ \ \ \ \ \ \ 
\frac{\partial v^i}{\partial\bar{x}^j}&=\frac{\partial\Phi^i_k}{\partial\bar{x}_j}(\bar{x})v^k \\
\frac{\partial x^i}{\partial\bar{v}^j}&=0 &\ \ \ \ \ \ \ \ 
\frac{\partial v^i}{\partial\bar{v}^j}&=\Phi^i_j(\bar{x})
\end{align}$$
Now we can actually compare metrics on $TM$ given by your definition. I'll refer to the basis of partial derivatives on $TTM$ as $e_i=\frac{\partial}{\partial x^i}$, $f_i=\frac{\partial}{\partial v^i}$ and likewise for $\bar{e}_i$, $\bar{f}_i$. Let $h=g_{ij}dx^idx^j+g_{ij}dv^idv^j$ and $\bar{h}=\bar{g}_{ij}d\bar{x}^id\bar{x}^j+\bar{g}_{ij}d\bar{v}^id\bar{v}^j$. If this definition is independent of coordinates, we must have $\psi^*h=\bar{h}$, where $\psi(\bar{x},\bar{v})=x,v$ is the transition function. We run into a problem on the $dx^idx^j$ components, though:
$$
(\psi^*h)_{ij}=(\psi^*h)(\bar{e}_i,\bar{e}_j)=h(\psi_*\bar{e}_i,\psi_*\bar{e}_j)
$$
$$
=h\left(\frac{\partial x^k}{\partial\bar{x}^i}e_k+\frac{\partial v^k}{\partial\bar{x}^i}f_k,\frac{\partial x^l}{\partial\bar{x}^j}e_l+\frac{\partial v^l}{\partial\bar{x}^j}f_l\right)=\Phi^k_i\Phi^l_jg_{kl}+\frac{\partial\Phi^k_m}{\partial\bar{x}^i}\frac{\partial\Phi^l_n}{\partial\bar{x}^j}v^m v^ng_{kl}
$$
We see we get an extra term related to the derivatives of the Jacobian. Conceptually, both this construction and the Sasaki metric use the fact that $T_vTM$ splits into vertical (tangent to fibers) and horizontal (transverse to fibers) subspaces. The vertical subspace inherits the metric from $TM$ in a straightforward way, but in the absence of the metric there isn't a canonical choice of horizontal subspace. Your construction uses the constant sections $(x,0)$ to choose a horizontal subspace, but this choice is coordinate dependent; the extra term in $\psi^*h$ is related to the degree to which the two coordinate systems do not agree on what a "constant" section is. The Sasaki metric  gets around this by making an intrinsic choice of horizontal subspace using the LC connection.
