Help with definition of Liouville measure

$$\require{AMScd}$$For a Riemannian manifold $$M$$, I have read authors talking about a 'Liouville measure' on the unit tangent bundle $$\operatorname{T}^1(M)$$ and then proceed to claim/prove that it is invariant under the geodesic flow. See for example Mautner's $$1957$$ paper 'Geodesic Flows on Symmetric Riemann Spaces.'

This measure is loosely defined as 'the measure on $$\operatorname{T}^1(M)$$ locally defined by $$\omega \wedge \theta$$ where $$\omega$$ is the volume form on $$M$$ associated to the metric and $$\theta$$ is an invariant volume form on the sphere' (again, see equation ($$17$$) of Mautner's paper).

I would like a precise definition/construction of such a measure on a general symmetric space. I have seen the example of the upper half-plane where the unit tangent bundle can be identified with a group and one uses the Haar measure.

Setup: Assume $$(G,\mathfrak{g})$$ is a non-compact, real, semi-simple, Lie group with finite center. Fix a Cartan involution $$\sigma: G \to G, d\sigma: \mathfrak{g} \to \mathfrak{g}$$ which then induces the decomposition $$\mathfrak{g}= \mathfrak{k}\oplus\mathfrak{p}$$ into the $$\pm 1$$ eigenspaces respectively. Moreover, if $$K = G_{\sigma}$$ (the fixed point set of $$\sigma$$), then $$\operatorname{Lie}(K)= \mathfrak{k}$$ and $$K$$ is compact since we assumed $$G$$ to have finite center. One can also check that $$\mathfrak{k}$$ and $$\mathfrak{p}$$ are invariant under $$\operatorname{Ad}(K)$$. By giving $$\mathfrak{p}$$ an $$\operatorname{Ad}(K)$$-invariant inner product, one can induce a $$G$$-invariant Riemannian metric on the tangent bundle of $$G/K$$. Additionally, $$G/K$$ has Riemannian geodesics of form $$t\mapsto g\exp(tX)K$$, with $$X\in \mathfrak{p}$$. Perhaps the example to keep in mind is $$\operatorname{SL}_n(\mathbb{R}) \simeq \exp(\mathfrak{p}) \times \operatorname{SO}(n)$$ where $$\mathfrak{p}$$ are symmetric, trace-zero matrices.

What I want: I would like to define a non-vanishing top form (Liouville measure) on $$\operatorname{T}^1(G/K)$$ which is 'locally of the type $$\omega \wedge \theta$$', where $$\omega$$ is the Riemannian volume form and $$\theta$$ is a non-vanishing top form on the sphere. Note that $$l_x$$, left multiplication by $$x\in G$$, is an isometry on $$G/K$$. And, of course, I would like the Liouville form to be invariant under the maps $$dl_x$$ so as to pass to the interesting class spaces $$\operatorname{T}^1(\Gamma\backslash(G/K)) \simeq \Gamma\backslash \operatorname{T}^1(G/K)$$, where $$\Gamma$$ is taken to be a freely acting lattice. Lastly, I would like to know if there is some sort of uniqueness statement that can be formulated.

An attempt: A naive attempt would be to say that $$\operatorname{T}^1(G/K) \simeq G/K \times S^d$$ is a trivial sphere-bundle since $$G/K\simeq \mathfrak{p}$$ is contractible. But I don't know how the action of $$G$$ would translate to $$G/K \times S^d$$.

Another attempt would be to use the commutative diagram

$$\begin{CD} G\times \mathfrak{p}_1 @>>> G \times_{K} \mathfrak{p}_1 \\ @V{\pi_1}VV @VV{\overline{q\circ\pi_1}}V \\ G @>{q}>> G/K \end{CD}$$

Here $$\mathfrak{p}_1$$ are the vectors in $$\mathfrak{p}$$ of norm $$1$$, $$K$$ acts on $$G\times \mathfrak{p}_1$$ on the right by $$(g,Y)\cdot k := (gk,\operatorname{Ad}(k^{-1})Y),$$ and $$G \times_{K} \mathfrak{p}_1$$ is shorthand for the quotient by this action. Note that $$G \times_{K} \mathfrak{p}_1$$ can be identified with $$\operatorname{T}^1(G/K)$$ by using the map $$(g,Y) \mapsto (g\exp(tY)K)'_{t=0}.$$ Under this identification, the action of $$dl_x: \operatorname{T}^1(G/K) \to \operatorname{T}^1(G/K)$$ for $$x\in G$$ is intertwined with the action $$x\cdot(g,Y)K := (xg,Y)K$$ on the first coordinate.

Thus it perhaps makes sense to try and construct a form $$\alpha$$ (of appropriate dimension) on $$G\times \mathfrak{p}_1$$ which is invariant (under both the left action of $$G$$ on the first component and the right action of $$K$$ specified above) and is also $$K$$-horizontal, that is to say that the interior product $$\iota_X \alpha =0$$ whenever $$X$$ is a vector field induced by the action of $$K$$. I guess such a form would induce a top form on $$G\times_K \mathfrak{p}_1$$ with the required properties?

Any help/references would be appreciated.

• The answer is in e.g. Besse, Manifolds all of whose geodesics are closed (1978, §1.M). As their 1.122 indicates, the invariant volume is the one associated to the contact structure of $T^1M$. – Francois Ziegler Jan 3 at 17:04
• If you are interested in symmetric spaces without compact components, then there is also an entirely geometric construction in terms of the geodesic currents (i.e., measures on the square of the visual boundary of the space). – R W Jan 3 at 20:58
• @FrancoisZiegler Oh, I get it. Would you like to put your comments down as an answer (copy paste is fine with me)? – Nope Jan 3 at 21:38
• @RW Yes, sounds interesting. Could you provide a reference/explanation please? – Nope Jan 3 at 21:39

The construction doesn’t really simplify on symmetric spaces. On $$TM\cong T^*M$$ (using the metric) consider the canonical 1-form $$\alpha=“\langle p,dq\rangle”$$ and symplectic form $$d\alpha$$ and hamiltonian vector field $$\xi$$ of $$H=\frac12\|p\|^2$$: $$\mathrm i_\xi d\alpha=-dH$$. Then $$\alpha$$ and $$\xi$$ restrict to a contact structure and its Reeb vector field on the level $$T^1M$$ of $$H$$: $$\mathrm i_\xi d\alpha = 0, \qquad\quad \mathrm i_\xi \alpha=1.$$ Moreover the geodesic flow is the Reeb flow. (For this Besse cites Weinstein (1974) who cites Berger (1965) who doesn’t cite Reeb (1950).) Now $$\mathrm L_\xi\alpha=\mathrm i_\xi d\alpha + d\mathrm i_\xi \alpha=0$$, so that flow preserves $$\alpha$$ and hence the volume form $$\alpha\wedge(d\alpha)^{\dim M-1}$$ of which Besse also gives a base $$\times$$ fiber description. Finally, any diffeo $$g$$ of $$M$$ lifts to a diffeo of $$T^*M$$ characterized by $$\langle g(p),Dg(q)(\delta q)\rangle$$ $$=$$ $$\langle p,\delta q\rangle$$, which “by construction” preserves $$\alpha$$. When $$g$$ is an isometry, it also preserves $$T^1M$$ $$\subset$$ $$TM$$ $$\cong$$ $$T^*M$$ and hence everything in sight.
Added: This “Lie” view of geodesics as produced by a contact flow (1896, pp. 96-102) works directly on $$\Gamma{\small\backslash} G/K$$; it may not have been that of Mautner, Gelfand-Fomin, or Hopf — they seem closer to the (of course equivalent) idea of putting a canonical (“Sasaki”) metric on $$TM$$ and $$T^1M$$ and using the resulting volume form, as in e.g. Paternain (1999, 1.17) or Berger (2003, pp. 195, 359, 472).
• No :-)$\,\!\,\!$ – Francois Ziegler Jan 5 at 16:13