Why is the Sasaki metric natural?

Question

Let $(M,g)$ be a Riemannian manifold with $\text{dim}(M)=n$. Then, there is a "natural" metric $\tilde{g}$ on the tangent bundle $TM$, so that $(TM,\tilde{g})$ is a Riemannian manifold, called the Sasaki metric, where a line element is written (with local coordinates of $TM$ given by $(x,v)$): $$ d\sigma^2 = g_{ij}\,dx^idx^j + g_{ij}\, Dv^iDv^j $$ where $D$ represents covariant differentiation: $$ Dv^i = dv^i + \Gamma_{jk}^iv^jdx^k $$ In components, letting indices range over $1$ to $n$, this is: \begin{align} \tilde{g}_{jk} &= g_{jk} + g_{\alpha \gamma}\Gamma_{\mu j}^\alpha\Gamma_{\eta k}^\gamma v^\mu v^\eta =: g_{jk}+A_{jk} \\ \tilde{g}_{j(n+k)} &= g_{kd}\Gamma^d_{\lambda j}v^\lambda =: B_{jk}\\ \tilde{g}_{(n+j)(n+k)} &= g_{jk} \end{align} Or, as a matrix: $$ \tilde{g} = \begin{bmatrix} g+A & B \\ B^T & g \end{bmatrix} $$

Question: intuitively speaking, why is this "natural"?

I am aware of other "natural" metrics on the tangent bundle; this question is specifically about this one, and the geometric intuition for why it is a good choice of metric. I can't seem to picture it.

Related: [1], [2], [3]

Answer 1

TL;DR: The connection gives us a way to canonically decompose $TTM$ as the direct sum of two copies of $TM$ (the "horizontal" and "vertical" bundles), so we just give each copy the Riemannian metric and declare the direct sum to be orthogonal. Long version:

A Riemannian metric on $TM$ is a (smoothly varying) choice of inner product on the double tangent space $T_v TM$ for each $v \in TM$. Since $\pi : TM \to M$ is a vector bundle over $M$, each $T_v TM$ has as a subspace the vertical tangent space $V_v TM$, which consists of the velocity vectors of curves in the vector space $T_{\pi(v)} M$, and thus can be canonically identified with $T_v M$. The Levi-Civita connection of $(M,g)$ provides a canonical horizontal subspace $H_v TM$, which consists of the velocity vectors of curves $(\gamma(t), V(t)) \in TM$ such that $v= (\gamma(0),V(0))$ and $\nabla_{\dot \gamma} V = 0.$

The upshot of all this is that we have a direct sum decomposition $TTM = VTM \oplus HTM$, with canonical isomorphisms $V_v TM \simeq T_{\pi(v)} M$ (described earlier) and $H_v TM \simeq T_{\pi(v)} M$ (by sending the velocity of $(\gamma,V)$ to the velocity of $\gamma$). If this isn't intuitive, think about the Euclidean case - if you have a tangent vector $v$ to $p= \pi(v) \in \mathbb R^n$, then the directions you can move it decouple in to one copy of $R^n$ for the motion of the basepoint and another copy for the motion of the vector.

The Sasaki metric can then be naturally defined by declaring $V_vTM$ and $H_vTM$ to be orthogonal, with the metric on each factor just being the pullback of $g$ from $T_{\pi(v)}M$ via the canonical isomorphisms.

This construction works for any vector bundle $E$ (over a Riemannian manifold $M$) equipped with a fibre metric and compatible connection: the vertical tangent spaces take the fibre metric from $E$, while the horizontal spaces (as defined by the connection) take the metric from $TM$. I have seen this general construction called the Kaluza-Klein metric.