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A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic line

The Alesker-Poincare pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cosphere-bundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally self-adjoint with respect to this pairing. As an application, the product structure of the space of SU(2)- and translation invariant valuations on the quaternionic line is described. The principal kinematic formula on the quaternionic line is stated and proved.


Smooth valuations on manifolds
Let M be a smooth manifold of dimension n. For simplicity, we suppose that M is oriented, although the whole theory works in the non-oriented case as well. Following Alesker, we set P(M) to be the set of compact submanifolds with corners. Definition 1.1. A valuation on M is a real valued map µ on P(M) which is additive in the following sense: whenever X, Y, X ∩ Y and X ∪ Y belong to P(M), then A set X ∈ P(M) admits a conormal cycle cnc(X), which is a compactly supported Legendrian cycle on the cosphere bundle S * M. Sometimes it will be convenient to think of S * M as the set of pairs (p, P ) with p ∈ M and P ⊂ T p M an oriented hyperplane, at other places it is better to think of it as the set of pairs (p, [ξ]) where p ∈ M and ξ ∈ T * p M \ {0} and the brackets denote the equivalence class for the relation ξ 1 ∼ ξ 2 ⇐⇒ ξ 1 = λξ 2 , λ > 0.
A valuation µ on M is called smooth if there exist an n − 1-form ω ∈ Ω n−1 (S * M) and an n-form φ ∈ Ω n (M) such that µ(X) = cnc(X)(ω) + X φ, X ∈ P(X). (1) If µ can be expressed in the form (1), we say that µ is represented by (ω, φ). The space of smooth valuations on M is denoted by V ∞ (M). It is a Fréchet space (see [6], Section 3.2 for the definition of the topology). If M = V is a vector space, the subspace of translation invariant smooth valuations will be denoted by Val sm (V ).
The valuation ρ * µ on N such that ρ * µ(X) := µ(ρ(X)), X ∈ P(N) is again smooth. If µ is represented by (ω, φ), then ρ * µ is represented by (ρ * ω, ρ * φ). This follows from the fact that cnc(ρ(X)) =ρ * cnc(X). Note also that ρ −1 = (ρ) −1 if ρ is a diffeomorphism. We will use some results of [11], which we would like to recall. The cosphere bundle S * M is a contact manifold of dimension 2n − 1 with a global contact form α (α is not unique, but this will play no role here). The projection from S * M to M will be denoted by π, it induces a linear map π * (fiber integration) on the level of forms.
The support of a smooth valuation µ is defined as The subspace of V ∞ (M) consisting of compactly supported valuations will be denoted by V ∞ c (M).

Let
: V ∞ c (M) → R denote the integration functional [7]. If µ has compact support, then µ := µ(X), where X ∈ P(M) is an ndimensional manifold with boundary containing spt µ in its interior. It is clear that, if µ is represented by (ω, φ) with compact supports, then Before stating our main theorem we have to recall two other constructions of Alesker.
The first one is the Euler-Verdier involution σ : V ∞ (M) → V ∞ (M) [6]. Let s : S * M → S * M be the natural involution on S * M, sending (p, P ) to (p,P ), whereP is the hyperplane P with the reversed orientation. If a valuation µ ∈ V ∞ (M) is represented by the pair (ω, φ), then σµ is defined as the valuation which is represented by the pair ((−1) n s * ω, (−1) n φ).
The second construction is the Alesker-Fu product [9], which is a bilinear map We refer to [9] for its construction. It is characterized by the following properties: (1) · is continuous and linear in both variables; (2) if ρ : N → M is a diffeomorphism and µ 1 , µ 2 ∈ V ∞ (M), then (3) if m 1 , m 2 are smooth measures on an n-dimensional vector space V , A 1 , A 2 ∈ K(V ) convex bodies with strictly convex smooth boundary and if µ i ∈ V ∞ (V ), i = 1, 2 is defined by then where ∆ : V → V × V is the diagonal embedding. Our first main theorem is the following relation between Alesker-Fu product, integration functional, Euler-Verdier involution and Rumin differential. Then Let us call the pairing the Alesker-Poincaré pairing. Note that Theorem 1.3 is equivalent to From Theorem 1.3 and from the fact that the Poincaré pairings on M and S * M are perfect, we get the following corollary (which was first proved by Alesker). Some more operators on V ∞ (M) were introduced in [11]. For this, we suppose that M is a Riemannian manifold. Then S * M admits an induced metric, the Sasaki metric [20].
If the smooth valuation µ is represented by (ω, φ), then Λµ is by definition the valuation which is represented by (L T ω + i T π * φ, 0).
Let us recall the definitions of the signature operator S and the Laplace operator ∆. Let * be the Hodge star acting on Ω * (S * M). Let µ ∈ V ∞ (M) be represented by (ω, φ). Then Sµ is defined as the valuation which is represented by ( * (Dω + π * φ), 0).
The Laplace operator ∆ is defined as ∆ := (−1) n S 2 . Our second main theorem shows that these operators fit well into Alesker's theory. In fact, they are formally self-adjoint with respect to the Alesker-Poincaré pairing.
, the following equations hold: We will apply these theorems in the study of the integral geometry of SU (2). This group acts on the quaternionic line H. In this setting, it is more natural to work with the space K(H) of convex sets instead of manifolds with corners. By Prop. 2.6. of [8], there is no loss of generality in doing so.
It was shown by Alesker [3] that the space of SU(2)-invariant and translation invariant valuations on the quaternionic line H is of dimension 10. For each purely complex number u of norm 1, let I u be the complex structure given by multiplication from the right with u and CP 1 u the corresponding Grassmannian of complex lines (with its unique SU(2)-invariant Haar measure). Alesker defined a valuation Z u by He showed that , together with Euler characteristic χ, the volume vol and the intrinsic volumes vol 1 , vol 3 form a basis of Val SU (2) . Following a suggestion of Fu, we will state the kinematic formula using a more symmetric choice. Noting that Z u = Z −u for all u ∈ S 2 , the 12 vertices ±u i , i = 1, . . . , 6 of an icosahedron on S 2 define 6 valuations Z u i , i = 1, . . . , 6.
We endow SU(2) with its Haar measure and the semidirect product SU(2) = SU(2) ⋉ H with the product measure. Let vol k denote the k-dimensional intrinsic volume [15]. Theorem 1.6. (Principal kinematic formula for SU(2)) Let K, L ∈ K(H). Then This theorem implies and generalizes the Poincaré formulas of Tasaki [19], (which contained an error in some constant) as we will explain in the last section.
1.0.1. Acknowledgements. I wish to thank Joseph Fu for illuminating discussions, in particular for the beautiful idea of using the vertices of an icosahedron as basis for Val

The Alesker-Poincaré pairing in terms of forms
In order to prove Theorem 1.3 and Theorem 1.4, we will need three lemmas which are of independent interest.
We represent χ by (ω, φ) and let µ i be the valuation represented by By inspecting the proof of Theorem 1.2 (which uses a local variational argument), one gets the following lemma.
Proof. We suppose M is non-compact (otherwise the statement is trivial). Let µ be represented by a pair (ω ′ , φ ′ ). Then . In other words, there is a compactly supported form ω ∈ H n−1 c (S * M) such that dω = Dω = Dω ′ +π * φ ′ −π * φ. By Theorem 1.2, the pair (ω, φ) represents µ up to a multiple of χ. Since the valuation represented by (ω, φ) and the valuation µ are both compactly supported, whereas χ is not, they have to be the same.
Next, both sides of (3) are linear in µ 1 and µ 2 . Using Lemma 2.1, we may therefore assume that the supports of µ 1 and µ 2 are contained in the support of a coordinate chart. Since the Alesker-Fu product, the Euler-Verdier involution and the integration functional are natural with respect to diffeomorphisms, it suffices to prove (3) in the case where M = V is a real vector space of dimension n.
Let us first suppose that µ 1 and σµ 2 are of the type (2). We thus have µ 1 (K) = m 1 (K + A 1 ) and σµ 2 (K) = m 2 (K + A 2 ) with smooth measures m 1 , m 2 and smooth convex bodies A 1 , A 2 with strictly convex boundary.
The left hand side of (3) is given by Let A ∈ K(V ) be smooth with strictly convex boundary. Its support function is defined by It is easy to show ( [10], [12]) that for X ∈ K(V ) We next compute that for all ( Let κ 2 ∈ Ω n (V ) be the form representing the measure m 2 . The first term in (10) By (11) we have (−1) n Ds * ω 2 + (−1) n π * φ 2 = G * A π * κ 2 . Applying s * to both sides and using (12), we get , which is the first term in (3).

Self-adjointness of natural operators
Proof of Theorem 1.5. Note first the following equation: This equation is immediate from (5) and the fact that s : S * M → S * M preserves orientation if and only if n is even.
Let µ i be represented by (ω i , φ i ). By Lemma 2.3 we may suppose that ω 2 and φ 2 are compactly supported.

The translation invariant case
From now on, V will denote an oriented n-dimensional real vector space. We will consider valuations on the space K(V ) of compact convex sets (i.e. convex valuations).
A convex valuation µ on V is called translation invariant, if µ(x + K) = µ(K) for all K ∈ K(V ) and all x ∈ V .
A translation invariant convex valuation µ is said to be of degree k if µ(tK) = t k µ(K) for t > 0 and K ∈ K(V ). By Val k (V ) we denote the space of translation invariant convex valuations of degree k. A valuation µ is even if µ(−K) = µ(K) and odd if µ(−K) = −µ(K), the corresponding spaces will be denoted by a superscript + or −.
In [16] it is shown that the space of translation invariant valuations can be written as a direct sum . The spaces Val 0 (V ) and Val n (V ) are both 1-dimensional (generated by χ and a Lebesgue measure respectively). For µ ∈ Val(V ), we denote by µ n its component of degree n.
Let us prove the following version of Theorem 1.3 in the translation invariant case. Theorem 4.1. Let µ 1 , µ 2 ∈ Val sm (V ) be represented by translation invariant forms (ω 1 , φ 1 ), (ω 2 , φ 2 ) respectively. Then (µ 1 · σµ 2 ) n is represented by the n-form Proof. The proof is similar to that of Theorem 1.3. Fix a Euclidean metric on V . For R > 0, let B R denote the ball of radius R, centered at the origin. Let us suppose that µ 1 (K) = vol(K + A 1 ) and σµ 2 (K) = vol(K + A 2 ) for all K ∈ K(V ). Then The first term is given by The second term yields Therefore we obtain This finishes the proof of Theorem 4.1 in the case where µ 1 , σµ 2 are of type K → vol(K + A). Using linearity of both sides, it also hold for linear combinations of such valuations. Since they are dense in Val(V ) (by Alesker's solution of McMullen's conjecture [1]), Theorem 4.1 is true in general.
Let us next suppose that V is endowed with a Euclidean product. We can identify Val n (V ) with R by sending vol to 1. We get a symmetric bilinear form (called Alesker pairing) Corollary 4.2. For valuations µ 1 , µ 2 ∈ Val sm (V ), the following equations hold: Proof. Analogous to the proof of Theorem 1.5.

Kinematic formulas and Poincaré formulas
5.1. Kinematic formulas. In this section, we suppose that G is a subgroup of O(V ) acting transitively on the unit sphere. By a result of Alesker [3], the space of translation invariant and G-invariant valuations Val G is a finite-dimensional vector space.
Let φ 1 , . . . , φ N a basis of Val G . Suppose we have a kinematic formula Here and in the following, G is endowed with its Haar measure and G := G ⋉ V with the product measure. Set
Fu [13] showed that these two maps are inverse to each other: For further use, we give another interpretation of (18). Let G be as above. The scalar product on the finite-dimensional space Val G induces a scalar product on Val G * such that PD is an isometry.

Klain functions.
Let us suppose additionally that −1 ∈ G, which implies that Val G ⊂ Val + . For 0 ≤ k ≤ n, the action of G on V induces an action on the Grassmannian Gr k (V ). We set P k := Gr k (V )/G for the quotient space. Given u ∈ P k , the space of k-planes contained in u admits a unique G-invariant probability measure and we define Z u ∈ Val G by Recall that the Klain function of an even, translation invariant valuation µ of degree k on a Euclidean vector space V is the function Kl µ : Gr k (V ) → R such that the restriction of µ to L ∈ Gr k (V ) is given by Kl µ (L) times the Lebesgue measure. An even, translation invariant valuation is uniquely determined by its Klain function [14]. If M is a compact k-dimensional submanifold (possibly with boundaries or corners), then Alesker proved the existence of a duality operator (or Fourier transform) F on Val +,sm such that Kl Fµ = Kl µ • ⊥ for all µ ∈ Val +,sm . F is formally self-adjoint with respect to the Alesker pairing.
Proof. Immediate from Lemma 2.2. of [12]. By G-invariance of φ i , we get For sufficiently close approximations of the m i by discrete measures Proof. Let u 1 , . . . , u N be such that Z u i , i = 1, . . . , N is a basis of Val G k . Let v 1 , . . . , v N be such that Z v j , j = 1, . . . , N is a basis of Val G n−k (note that the dimensions of these two spaces agree by Thm. 1.2.2 in [2]).
Setting g ij := Z u i , Z v j and (g ij ) for the inverse matrix, the principal kinematic formula implies that for all M 1 and M 2 as above This shows that where the last equation follows from (19) and the self-adjointness of F.
A canonical representative in the preimage of (a : b : c) ∈ RP 2 is given by the 2-plane spanned by 1 and ai + bj + ck.
If u = (a, b, c) ∈ S 2 , then the planes in u are the complex lines for the complex structure I u which is defined by multiplication by u from the right on H. We will therefore write CP u instead of u. Note that CP u = CP −u .
The following SU(2)-invariant and translation invariant valuations of degree 2 were introduced by Alesker [3]. Proof of Theorem 1.6. From Lemma 5.3 we infer that there is a finite number of points u 1 , . . . , u N ∈ RP 2 such that Z u i , i = 1, . . . , N form a basis of Val SU (2) 2 . Alesker showed that N = 6 and that is such a basis.
Our aim is to compute the product Z u , Z v for u, v ∈ RP 2 . We will achieve it by first expressing each Z u as a smooth valuation represented by some 3-form ω u ∈ Ω 3 (S * H) and then applying Theorem 4.1.
Since the metric induces a diffeomorphism between S * H and SH, we may as well work with the latter space. The image of the conormal cycle of a compact convex set K under this diffeomorphism is the normal cycle nc(K).
Let us introduce several differential forms on SH, depending on the choice of the complex structure I u . We follow the notation of [17].
Let α, β, γ be 1-forms on SH which, at a point (x, v) ∈ SH, equal Note that α is the canonical 1-form (in particular independent of u), whereas β u and γ u depend on u.
Let Ω be the pull-back of the symplectic form on (H, I u ) to SH, i.e.
Claim: Z u is represented by the 3-form Since ω u is U(2)-and translation invariant and has bidegree (2, 1) (with respect to the product decomposition SH = H × S(H)), it represents some U(2)-invariant, translation invariant valuation µ u of degree 2. Here U(2) is the unitary group for the complex structure I u .
Now the space of valuations with these properties is of dimension 2 [2]. It is thus enough to show that the valuation Z u and the valuation µ u agree on the unit ball B as well as on a complex disk D u .
It is clear that Z u (B) = ω 2 = π. It was shown by Fu (compare Equation (37) in [13],) that Z u (D u ) = π 2 . By [11], the derivation of a smooth translation invariant valuation µ on a finite-dimensional Euclidean vector space is given by It follows that, if µ is of degree k, then Λµ(B) = kµ(B).
Let ±u i , i = 1, . . . , 6 be the 12 vertices of an icosahedron I on S 2 . They induce 6 valuations Z u i , i = 1, . . . , 6. Since the edge length a of I satisfies cos a =