### Extensions of (co)vector fields to tangent bundles

I am reading Sasaki’s original paper on the construction of the Sasaki metric (a canonical Riemannian metric on the tangent bundle of a Riemannian manifold), and the following took me way too long to understand. So I’ll write it down in case I forgot in the future.

In section two of the paper, Sasaki consider “extended transformations and extended tensors”. Basically he wanted to give a way to “lift” tensor fields from a manifold to tensor fields of the same rank on its tangent bundle. And he did so in the language of coordinate changes, which geometrical content is a bit hard to parse. I’ll discuss his construction in a bit. But first I’ll talk about something different.

The trivial lifts
Let $M, N$ be smooth manifolds, and let $f:M\to N$ a submersion. Then we can trivially lift covariant objects on $N$ to equivalent objects on $M$ by the pull-back operation. To define the pull-back, we start with a covariant tensor field $\vartheta \in \Gamma T^0_kN$, and set $f^*\vartheta \in \Gamma T^0_kM$ by the formula:

$\displaystyle f^*\vartheta(X_1,\ldots,X_k) = \vartheta(df\circ X_1, \ldots, df\circ X_k)$

where the $X_1, \ldots, X_k \in T_pM$, and we use that $df(p): T_pM \to T_{f(p)}N$. Observe that for a function $g: N \to \mathbb{R}$, the pull-back is simply $f^*g = g\circ f :M\to N\to\mathbb{R}$.

On the other hand, for contravariant tensor fields, the pull-back is not uniquely defined: using that $f$ is a submersion, we have that $TM / \ker(df) = TN$, so while, given a vector field $v$ on $N$, we can always find a vector field $w$ on $M$ such that $df(w) = v$, the vector field $w$ is only unique up to an addition of a vector field that lies in the kernel of $df$. If, however, that $M$ is Riemannian, then we can take the orthogonal decomposition of $TM$ into the kernel and its complement, thereby getting a well-defined lift of the vector field (in other words, by exploiting the identification between the tangent and cotangent spaces).

Remarkably, the extensions defined by Sasaki is not this one.

(Let me just add a remark here: given two manifolds, once one obtain a well defined way of lifting vectors, covectors, and functions from one to the other, such that they are compatible ($\vartheta^*(v^*) = [\vartheta(v)]^*$), one can extend this mapping to arbitrary tensor fields.)

The extensions defined by Sasaki
As seen above, if we just rely on the canonical submersion $\pi:TM\to M$, we cannot generally extend vector fields. Sasaki’s construction, however, strongly exploits the fact that $TM$ is the tangent bundle of $M$.

We start by looking at the vector field extension defined by equation (2.6) of the linked paper. We first observe that a vector field $v$ on a manifold $M$ is a section of the tangent bundle. That is, $v$ is a map $M\to TM$ such that the composition with the canonical projection $\pi\circ v:M\to M$ is the identity map. This implies, using the chain rule, that the map $d(\pi\circ v)= d\pi \circ dv: TM\to TM$ is also the identity map. Now, $d\pi: T(TM) \to TM$ is the projection induced by the projection map $\pi$, which is different from the canonical projection $\pi_2: T(TM) \to TM$ from the tangent bundle of a manifold to the manifold itself. However, a Proposition of Kobayashi (see “Theory of Connections” (1957), Proposition 1.4), shows that there exists an automorphism $\alpha:T(TM) \to T(TM)$ such that $d\pi \circ \alpha = \pi_2$ and $\pi_2\circ\alpha = d\pi$. So $v$ as a differential mapping induces a map $\alpha\circ dv: TM \to T(TM)$, which is a map from the tangent bundle $TM$ to the double tangent bundle $T(TM)$, which when composed with the canonical projection $\pi_2$ is the identity. In other words, $\alpha\circ dv$ is a vector field on $TM$.

Next we look at the definition (2.7) for one-forms. Give $\vartheta$ a one-form on $M$, it induces naturally a scalar function on $TM$: for $p\in M, v\in T_pM$, we have $\vartheta: TM\to \mathbb{R}$ taking value $\vartheta(p)\cdot v$. Hence its differential $d\vartheta$ is a one-form over $TM$.

Now, what about scalar functions? Let $\vartheta$ be a one-form and $v$ be a vector field on $M$, we consider the pairing of their extensions to $TM$. It is not too hard to check that the corresponding scalar field to $\vartheta(v)$, when evaluated at $(p,w)\in TM$, is in fact $d(\vartheta(v))|_{p,w}$, the derivative of the scalar function $\vartheta(v)$ in the direction of $w$ at point $p$. In general, the compatible lift of scalar fields $g:M\to \mathbb{R}$ to $TM$ is the function $\tilde{g}(p,v) = dg(p)[v]$.

Using this we can extend the construction to arbitrary tensor fields, and a simple computation yields that this construction is in fact identical, for rank-2 tensors, to the expressions given in (2.8), (2.9), and (2.10) in the paper.

The second extension
The above extension is not the only map sending vectors on $M$ to vectors on $TM$. In the statement of Lemmas 3 there is also another construction. Given a vector field $v$, it induces a one parameter family of diffeomorphisms on $TM$ via that maps $\psi_t(p,w) = (p, w+vt)$. Its differential $\frac{d}{dt}\psi_t|_{t=0}$ is a vector field over $TM$.

The construction in the statement of Lemma 4 is the trivial one mentioned at the start of this post.