Rigorous Treatment of Flow Lifting to Tangent and Cotangent Bundles
Preliminaries
Let $M$ be a smooth manifold of dimension $n$. We denote by:
- $TM$ the tangent bundle of $M$, with projection $\pi: TM \to M$
- $T^M$ the cotangent bundle of $M$, with projection $\pi^: T^*M \to M$
- $\mathfrak{X}(M)$ the space of smooth vector fields on $M$
- $\Omega1(M)$ the space of smooth 1-forms on $M$
- $\text{Diff}(M)$ the group of diffeomorphisms of $M$
Section 1: Lifting to the Tangent Bundle: Pushforward Construction
1.1 The Tangent Functor
Definition 1.1 (Tangent Functor).
The tangent functor $T$ assigns to each smooth manifold $M$ its tangent bundle $TM$, and to each smooth map $f: M \to N$ the pushforward (or differential)
$$
Tf \equiv f: TM \to TN, \quad v \mapsto f(v)
$$
defined by $f(v)(g) = v(g \circ f)$ for $g \in C\infty(N)$. For $v \in T_pM$, we have $f(v) \in T_{f(p)}N$.
Lemma 1.2 (Properties of the Pushforward).
For smooth maps $f: M \to N$ and $g: N \to P$, we have:
- $(g \circ f)* = g* \circ f_*$ (functoriality)
- $(\text{id}M)* = \text{id}_{TM}$
- If $f$ is a diffeomorphism, then $f: TM \to TN$ is a diffeomorphism with $(f)^{-1} = (f^{-1})_*$
1.2 Canonical Lifting to $TM$
Definition 1.3 (Lifted Flow on $TM$).
Given a flow $\Phi_t: M \to M$ generated by $V \in \mathfrak{X}(M)$, the lifted flow on $TM$ is defined by applying the tangent functor:
\[
\tilde{\Phi}t := T\Phi_t = (\Phi_t): TM \to TM
\]
Explicitly, for $v \in T_pM$:
\[
\tilde{\Phi}t(v) = (\Phi_t)(v) = d\Phi_t|p(v) \in T{\Phi_t(p)}M
\]
Theorem 1.4 (Lifted Flow Properties).
The family $\{\tilde{\Phi}t\}{t\in\mathbb{R}}$ satisfies:
- $\tilde{\Phi}0 = \text{id}{TM}$
- $\tilde{\Phi}{t+s} = \tilde{\Phi}t \circ \tilde{\Phi}_s$ for all $t,s \in \mathbb{R}$
- $\pi \circ \tilde{\Phi}_t = \Phi_t \circ \pi$ (bundle morphism property)
- $\tilde{\Phi}_t$ is a diffeomorphism of $TM$ for each $t \in \mathbb{R}$
Thus $\tilde{\Phi}_t$ is a genuine flow on the manifold $TM$.
Proof.
(1) $\tilde{\Phi}0 = T\Phi_0 = T(\text{id}M) = \text{id}_{TM}$.
(2) By functoriality:
\[
\tilde{\Phi}{t+s} = T(\Phi{t+s}) = T(\Phi_t \circ \Phi_s) = T\Phi_t \circ T\Phi_s = \tilde{\Phi}t \circ \tilde{\Phi}s
\]
(3) For $v \in T_pM$:
\[
\pi(\tilde{\Phi}t(v)) = \pi(d\Phi_t|p(v)) = \Phi_t(p) = \Phi_t(\pi(v))
\]
(4) Since $\Phi_t$ is a diffeomorphism, $T\Phi_t = (\Phi_t)$ is a diffeomorphism with inverse $T\Phi{-t} = (\Phi_t)_^{-1}$. ∎
Proposition 1.5 (Commutative Diagram).
The lifting construction yields the following commutative diagram:
\[
\begin{array}{ccc}
TM & \xrightarrow{\tilde{\Phi}t = (\Phi_t)*} & TM \\
\pi\downarrow & & \downarrow\pi \\
M & \xrightarrow{\quad\Phi_t\quad} & M
\end{array}
\]
This makes $\tilde{\Phi}_t$ a bundle automorphism covering $\Phi_t$.
1.3 The Generating Vector Field: Complete Lift
Definition 1.6 (Complete Lift).
For $V \in \mathfrak{X}(M)$ generating $\Phi_t$, the complete lift $VC \in \mathfrak{X}(TM)$ is defined by:
\[
VC_v := \left.\frac{d}{dt}\right|{t=0} \tilde{\Phi}t(v) = \left.\frac{d}{dt}\right|_{t=0} d\Phi_t(v)
\]
for $v \in TM$.
Theorem 1.7 (Coordinate Expression).
In local coordinates $(xi)$ on $M$ and induced coordinates $(xi, vi)$ on $TM$, if $V = Vi(x)\frac{\partial}{\partial xi}$, then:
\[
VC = Vi(x)\frac{\partial}{\partial xi} + \frac{\partial Vj}{\partial xi}(x)vi\frac{\partial}{\partial vj}
\]
Proof.
Let $\Phi_t(x) = (\Phi_t1(x), \ldots, \Phi_tn(x))$. Then:
\[
d\Phi_t\left(\frac{\partial}{\partial xi}\right) = \frac{\partial \Phi_tj}{\partial xi}\frac{\partial}{\partial xj}
\]
For a vector $v = vi\frac{\partial}{\partial xi}$, we have:
\[
\tilde{\Phi}t(v) = vi\frac{\partial \Phi_tj}{\partial xi}\frac{\partial}{\partial xj}
\]
Differentiating at $t=0$:
\[
VC_v = \left.\frac{d}{dt}\right|{t=0} \left(vi\frac{\partial \Phi_tj}{\partial xi}\right)\frac{\partial}{\partial xj}
= vi\frac{\partial Vj}{\partial xi}\frac{\partial}{\partial xj}
\]
In coordinates $(xi, vi)$ on $TM$, this becomes the stated expression. ∎
Definition 1.8 (Horizontal and Vertical Components).
The complete lift decomposes as:
\[
VC = VH + VV
\]
where:
- $VH = Vi(x)\frac{\partial}{\partial xi}$ is the horizontal lift (base motion)
- $VV = \frac{\partial Vj}{\partial xi}vi\frac{\partial}{\partial vj}$ is the vertical component (fiber linearization)
Section 2: Lifting to the Cotangent Bundle: Pullback Construction
2.1 The Cotangent Functor
Definition 2.1 (Cotangent Functor).
The cotangent functor $T^$ assigns to each smooth manifold $M$ its cotangent bundle $T^M$, and to each diffeomorphism $f: M \to N$ the pullback
\[
T^f \equiv f^: T^N \to T^M, \quad \alpha \mapsto f^\alpha
\]
defined by $(f^\alpha)p(v) = \alpha{f(p)}(f_*(v))$ for $v \in T_pM$.
Lemma 2.2 (Properties of the Pullback).
For diffeomorphisms $f: M \to N$ and $g: N \to P$, we have:
- $(g \circ f)^* = f^* \circ g^*$
- $(\text{id}M)^* = \text{id}{T^*M}$
- $(f^)^{-1} = (f^{-1})^$
2.2 Canonical Lifting to $T^*M$
Definition 2.3 (Lifted Flow on $T^M$).
Given a flow $\Phi_t: M \to M$ generated by $V \in \mathfrak{X}(M)$, the lifted flow on $T^M$ is defined by the pullback:
\[
\hat{\Phi}t := (\Phi_t)^: T^M \to T^M
\]
Explicitly, for $\alpha \in T^pM$:
\[
\hat{\Phi}t(\alpha) = (\Phi_t)^(\alpha) \in T^{\Phi_t^{-1}(p)}M
\]
with the action: $\hat{\Phi}t(\alpha)(v) = \alpha((\Phi_t)*(v))$ for $v \in T_{\Phi_t^{-1}(p)}M$.
Theorem 2.4 (Lifted Flow Properties on $T^*M$).
The family $\{\hat{\Phi}t\}{t\in\mathbb{R}}$ satisfies:
- $\hat{\Phi}0 = \text{id}{T^*M}$
- $\hat{\Phi}{t+s} = \hat{\Phi}t \circ \hat{\Phi}_s$ for all $t,s \in \mathbb{R}$
- $\pi^* \circ \hat{\Phi}_t = \Phi_t^{-1} \circ \pi^*$ (bundle morphism property)
- $\hat{\Phi}_t$ is a diffeomorphism of $T^*M$ for each $t \in \mathbb{R}$
Proof.
(1) $\hat{\Phi}0 = (\Phi_0)^* = (\text{id}M)^* = \text{id}_{T^*M}$.
(2) Using Lemma 2.2(1):
\[
\hat{\Phi}{t+s} = (\Phi{t+s})^* = (\Phi_t \circ \Phi_s)^* = \Phi_s^* \circ \Phi_t^* = \hat{\Phi}s \circ \hat{\Phi}t
\]
Note the order reversal due to contravariance.
(3) For $\alpha \in T^pM$, we have $\pi^(\hat{\Phi}t(\alpha)) = \pi^((\Phi_t)^\alpha) = \Phi_t^{-1}(p) = \Phi_t^{-1}(\pi^*(\alpha))$.
(4) Since $\Phi_t$ is a diffeomorphism, $(\Phi_t)^$ is a diffeomorphism with inverse $(\Phi_t^{-1})^ = (\Phi_{-t})^*$. ∎
Proposition 2.5 (Commutative Diagram for Cotangent Lift).
The lifting construction yields:
\[
\begin{array}{ccc}
T^M & \xrightarrow{\hat{\Phi}_t = (\Phi_t)^} & T^M \\
\pi^\downarrow & & \downarrow\pi^* \\
M & \xrightarrow{\quad\Phi_t^{-1}\quad} & M
\end{array}
\]
2.3 The Generating Vector Field: Cotangent Lift
Definition 2.6 (Cotangent Lift).
For $V \in \mathfrak{X}(M)$ generating $\Phi_t$, the cotangent lift $V^* \in \mathfrak{X}(T^M)$ is defined by:
\[
V^\alpha := \left.\frac{d}{dt}\right|{t=0} \hat{\Phi}t(\alpha) = \left.\frac{d}{dt}\right|{t=0} (\Phi_t)^\alpha
\]
for $\alpha \in T^M$.
Theorem 2.7 (Coordinate Expression).
In local coordinates $(xi)$ on $M$ and induced coordinates $(xi, p_i)$ on $T^M$, if $V = Vi(x)\frac{\partial}{\partial xi}$, then:
\[
V^ = Vi(x)\frac{\partial}{\partial xi} - \frac{\partial Vj}{\partial xi}(x)p_j\frac{\partial}{\partial p_i}
\]
Proof.
Let $\alpha = p_i dxi$. Under the flow $\Phi_t$, coordinates transform as $xi \mapsto \Phi_ti(x)$. The pullback acts on forms as:
\[
(\Phi_t)^(p_i dxi) = p_i \frac{\partial \Phi_ti}{\partial xj} dxj
\]
Differentiating at $t=0$:
\[
\left.\frac{d}{dt}\right|_{t=0} (\Phi_t)^\alpha = p_i \frac{\partial Vi}{\partial xj} dxj
\]
In Hamiltonian vector field form, this yields the stated expression. ∎
Definition 2.8 (Hamiltonian Vector Field).
For a smooth function $H: T^M \to \mathbb{R}$, define the Hamiltonian vector field $X_H$ on $T^M$ by:
\[
\iota_{X_H}\omega = dH
\]
where $\omega = dp_i \wedge dxi$ is the canonical symplectic form. Then $V^*$ can be identified with $X_H$ for $H(x,p) = p_i Vi(x)$.
Section 3: Functorial Properties and Naturality
Theorem 3.1 (Functoriality).
The lifting constructions define functors:
- Tangent functor: $T: \text{Diff}(M) \to \text{Diff}(TM)$, $\Phi \mapsto \Phi_*$
- Cotangent functor: $T^: \text{Diff}(M) \to \text{Diff}(T^M)$, $\Phi \mapsto \Phi^*$
satisfying:
\[
T(\Phi \circ \Psi) = T\Phi \circ T\Psi, \quad T^(\Phi \circ \Psi) = T^\Psi \circ T^*\Phi
\]
Theorem 3.2 (Naturality with Respect to Vector Fields).
Let $V, W \in \mathfrak{X}(M)$ with flows $\Phi_tV, \Phi_tW$. Then:
- The complete lifts satisfy: $[VC, WC] = [V,W]C$
- The cotangent lifts satisfy: $[V^, W^] = [V,W]^*$
Proposition 3.3 (Relation to Lie Derivative).
For $\alpha \in \Omega1(M)$ and $V \in \mathfrak{X}(M)$:
\[
\left.\frac{d}{dt}\right|{t=0} (\Phi_t)^*\alpha = \mathcal{L}V\alpha
\]
where $\mathcal{L}_V$ is the Lie derivative.
Similarly, for $X \in \mathfrak{X}(M)$:
\[
\left.\frac{d}{dt}\right|{t=0} (\Phi_t)*X = \mathcal{L}_VX
\]
Section 4: Summary and Comparison
Theorem 4.1 (Summary of Lifting Constructions).
| Property | Tangent Bundle ($TM$) | Cotangent Bundle ($T^M$) |
|---|
| Functor | Tangent functor $T$ | Cotangent functor $T^*$ |
| Action | Pushforward $(\Phi_t)$ | Pullback $(\Phi_t)^*$ |
| Flow | $\tilde{\Phi}_t = T\Phi_t$ | $\hat{\Phi}t = T^\Phi_t$ |
| Generator | Complete lift $VC$ | Cotangent lift $V^*$ |
| Coordinates | $VC = Vi\partial_{xi} + (\partial_i Vj)vi\partial{vj}$ | $V^* = Vi\partial_{xi} - (\partial_i Vj)p_j\partial{p_i}$ |
| Diagram | $\pi \circ \tilde{\Phi}_t = \Phi_t \circ \pi$ | $\pi^* \circ \hat{\Phi}_t = \Phi_t^{-1} \circ \pi^*$ |
| Covariance | Covariant functor | Contravariant functor |
Corollary 4.2 (Naturality of Constructions).
Both lifting constructions are canonical (require no additional structure) and functorial, making them the natural ways to extend flows from the base manifold to its tangent and cotangent bundles.
Remark 4.3 (Physical Interpretation).
- Tangent bundle lift: Describes how velocity vectors are transported by the flow
- Cotangent bundle lift: Describes how momentum/force covectors are transported by the flow
- The sign difference in the vertical components reflects the dual nature of vectors and covectors under coordinate transformations
Section 5: Applications
Definition 5.1 (Natural Lift of Tensor Fields).
For any tensor field $T$ of type $(r,s)$ on $M$, define its natural lift under $\Phi_t$ by:
\[
(\Phi_t)T := (\Phi_t)^{\otimes r} \otimes (\Phi_t)^{*\otimes s} T
\]
Theorem 5.2 (Infinitesimal Generator).
The infinitesimal generator of the lifted flow on tensor fields is the Lie derivative:
\[
\left.\frac{d}{dt}\right|{t=0} (\Phi_t)*T = \mathcal{L}_VT
\]
Application 5.3 (Symplectic Geometry).
On $T^M$ with canonical symplectic form $\omega = d\theta$ (where $\theta = p_i dxi$ is the tautological 1-form), we have:
\[
\mathcal{L}{V^}\omega = 0 \quad \text{and} \quad \mathcal{L}{V^}\theta = 0
\]
Thus $V^$ is a Hamiltonian vector field preserving the symplectic structure.
Application 5.4 (Geometric Mechanics).
If $\Phi_t$ is a symmetry of a Lagrangian $
