IIntrinsic volumes of sublevel sets
Benoît JubinApril 8, 2020
Abstract
We establish formulas that give the intrinsic volumes, or curvature measures, ofsublevel sets of functions defined on Riemannian manifolds as integrals of functionalsof the function and its derivatives. For instance, in the Euclidean case, if f ∈ C ( R n , R )and 0 is a regular value of f , then the intrinsic volume of degree n − k of the sublevelset M f := f − (] −∞ , L n − k ( M f ) = Γ( k/ π k/ ( k − Z M div P n,k (Hess( f ) , ∇ f ) q f k − + k∇ f k k − ∇ f vol n for 1 ≤ k ≤ n , where the P n,k ’s are polynomials given in the text.This includes as special cases the Euler–Poincaré characteristic of sublevel setsand the nodal volumes of functions defined on Riemannian manifolds. Therefore,these formulas give what can be seen as generalizations of the Kac–Rice formula.Finally, we use these formulas to prove the Lipschitz continuity of the intrinsicvolumes of sublevel sets. Contents
Introduction 11 Intrinsic volumes 32 Intrinsic volumes of sublevel sets 5
Introduction
Intrinsic volumes are geometric invariants attached to well-behaved subsets of Riemannianmanifolds. They include the volume and the Euler–Poincaré characteristic. Among theirapplications in the field of integral geometry are Weyl’s tube formula ([21]), that givesthe volumes of tubular neighborhoods of submanifolds, and the kinematic formula ofBlaschke, Chern and Santaló ([8, 9]), that gives the volume of the Minkowski sum of
Keywords: intrinsic volume, curvature measure, Lipschitz–Killing curvature, Euler–Poincaré charac-teristic, sublevel set, excursion set, nodal set, nodal volume, Kac–Rice formula. a r X i v : . [ m a t h . DG ] A p r wo convex bodies. They were introduced in their modern form by Herbert Federer inthe seminal article [10], where they are called curvature measures, after special cases inconvex geometry were treated by Hermann Minkowski. Among the vast literature ontheir subject, we only mention the book [17], the survey on a related topic [20], and thearticles [12, 13, 23].In this article, we study the intrinsic volumes of sublevel sets of functions definedon Riemannian manifolds. These were already studied from the point of view of Morsetheory in [11]. Since intrinsic volumes include the volume of the boundary, this studyencompasses volumes of level sets, and in particular of zero sets, also called nodal sets.The first closed explicit formulas computing nodal volumes appeared in [3], which was amotivation for the present article. These formulas can be seen as generalizations of theso-called Kac–Rice formula (see for instance [18]).Sublevel sets are also studied in probability theory, where superlevel sets of randomfields are called excursion sets; see for instance the books [1, 2] and the articles [3, 16, 22].The importance of the formulas obtained in this paper for the study of random fields (asstudied in [3]), compared to existing Kac–Rice formulas, stems from the fact that they arein “closed form” as opposed to being limits of integrals depending on a parameter. Main results
We now describe the contents of this article in more detail. In thisintroduction, we restrict ourselves to the flat case. In Section 1, we recall the definitionand main properties of intrinsic volumes. If N is a flat compact n -dimensional Riemannianmanifold with boundary, they take the form L n − k ( N ) = b k Z ∂N tr (cid:18)^ k − S (cid:19) vol ∂N (1)for 0 ≤ k ≤ n , where b k ∈ R and S is the second fundamental form of ∂N in N .In Section 2, we specialize our study to the case where N is a sublevel set. Namely,let M be a flat n -dimensional Riemannian manifold (without boundary), let f ∈ C ( M, R ),and assume that a ∈ R is a regular value of f and that the sublevel set M a := f − (] −∞ , a ])is compact. The second fundamental form of ∂M a in M can be expressed in terms of thegradient and the Hessian of f . An important lemma (Lemma 2.2) establishes that theabove integrand is then a polynomial in ∇ f and Hess( f ) divided by k∇ f k k − . We thenuse the divergence theorem to transform the above integral over ∂M a into an integral over M a . This leads to our main formula which, in the flat case, reads L n − k ( M a ) = b k Z M a div P n,k (Hess( f ) , ∇ f ) q ( f − a ) k − + k∇ f k k − ∇ f vol M (2)for 1 ≤ k ≤ n , where the P n,k ’s are polynomials given in the text (Theorem 2.9). Themain advantage of this formula is that it is an explicit integral over M a (and not ∂M a )of a continuous functional in f and its derivatives up to order 3.Since the intrinsic volume of degree n − M is compact, one can use the intrinsic volume of either the sublevel or the superlevel set,yielding for the volume of the zero set Z f of f the formulavol( Z f ) = 12 Z M σ f η f (cid:16) f k∇ f k + Hess( f )( ∇ f, ∇ f ) − η f ∆ f (cid:17) vol M (3)where σ f is the sign of f and η f := q f + k∇ f k . This formula was obtained in thecase of a flat torus in [3]. As in [3], one can do an integration by parts to eliminate σ f M where the integrand is a Lipschitz continuous functional of f and its derivatives up to order 2 (Equation (49)). This regularity allows one to applytechniques of the Malliavin calculus to obtain results about the expected value, varianceand higher moments of the nodal volumes of certain families of random fields (see [3]),and more generally of the intrinsic volumes of their excursion sets.In Section 3, after recalling basic facts on natural topologies on C p ( M, R ), the uniformand the (Whitney) strong C p -topologies, we prove that conditions needed to establish ourformulas (regularity of the value and compactness of the sublevel sets) are generic. Then,we prove the continuity of intrinsic volumes of sublevel sets. For instance, if 0 ≤ k ≤ n ,then the function L sub n − k : Reg ( M, R ) U −→ R (4)( f, a ) n − k ( M af )is Lipschitz continuous, where the domain is the set of couples ( f, a ) where f ∈ C ( M, R )is proper bounded below and a ∈ R is a regular value of f , and is equipped with theuniform C -topology (Theorem 3.10). In particular, the Euler–Poincaré characteristic ofsublevel sets is locally constant. Conventions and notation • If P is a proposition, then [ P ] := 1 if P else 0. • We denote by pr i the projection on the i th factor of a direct product. • The bracket b−c : R → Z denotes the floor function. • If a, b ∈ R , then (cid:74) a, b (cid:75) := Z ∩ [ a, b ]. • If a ∈ R , then N ≥ a := { n ∈ N | n ≥ a } ∪ {∞} and similarly for similar symbols. • The symbol J (resp. V ) denotes the symmetric (resp. exterior) product or power ofvector spaces. • Unless otherwise specified, manifolds are Hausdorff, paracompact, real, finite-dimensional,and smooth, that is, of class C ∞ . • The space of smooth sections of the vector bundle E → M is denoted by Γ( E → M )or Γ ( p ) ( E → M ) if the differentiability class p ∈ N need be specified. For instance,the metric tensor of a Riemannian manifold M is an element of Γ( J T ∗ M → M ). Let (
M, g ) be an n -dimensional compact Riemannian manifold with boundary. Its metricwill also be denoted by h− , −i and the associated norm by k−k . We denote by ∇ lc its Levi-Civita connection. Let vol M be the Riemannian density on M and vol ∂M be the induceddensity on ∂M . The symbol vol will also denote the associated volume of (sub)manifolds.Let R ∈ Γ( J V T ∗ M → M ) be the covariant curvature tensor of M . Let S :=(( ∇ lc ν | T ∂M ) T ) [ ∈ Γ( J T ∗ ∂M → ∂M ) be the second fundamental form of ∂M in M ,where ν ∈ Γ( T M | ∂M → ∂M ) is the outward unit normal vectorfield on ∂M and ( − ) T : T M | ∂M → T ∂M denotes the tangential component, and [ : T ∂M → T ∗ ∂M denotes the musical iso-morphism induced by the metric. The symbol tr denotes the trace of a bilinear form.For the exterior product of symmetric bilinear forms, ∧ : J V p V × J V q V → J V p + q V , also called in differential geometry the Kulkarni–Nomizu product, see forinstance [10, §2]. 3or 0 ≤ k ≤ n , the intrinsic volume of degree n − k of M is defined as L n − k ( M ) := a k Z M tr (cid:18)^ k/ R (cid:19) vol M + b k − c X m =0 b k,m Z ∂M tr (cid:18)^ m R | ∂M ∧ ^ k − − m S (cid:19) vol ∂M (5)where a k := [ k even]( − π ) k/ ( k/ ≤ k, (6) b k,m := ( − m Γ( k/ − m )2 m +1 π k/ m !( k − − m )! for 0 ≤ m ≤ (cid:22) k − (cid:23) . (7)We also set b k := b k, = Γ( k/ π k/ ( k − ≤ k. (8) Remark . Formula (5) was obtained by specializing the general definition [1, Def. 10.7.2],which holds for Whitney stratified spaces “of positive reach” in Riemannian manifolds (in-cluding Riemannian manifolds with corners), to the case of Riemannian manifolds withboundary. The integrands, which are contractions of the curvature tensor and the secondfundamental form, are called the
Lipschitz–Killing curvatures of ∂M in M . Moregeneral versions L n − k ( M, A ) can be defined for Borel subsets A ⊆ M and are called cur-vature measures in M . The intrinsic volumes are the total measures of these curvaturemeasures, that is, L n − k ( M ) = L n − k ( M, M ).One has L n ( M ) = vol( M ) , (9) L n − ( M ) = 12 vol( ∂M ) , (10) L ( M ) = χ ( M ) . (11)The first two equalities immediately follow from a = 1 and from a = 0 and b = respectively (indeed, tr( V R x ) is the trace of the identity of V T ∗ x M ’ R , which is 1, andsimilarly for tr( V S x )). The third equality is the Gauss–Bonnet–Chern theorem (see [6, 7]for the original articles, and [19] for manifolds with boundary), where the right-hand sideis the Euler–Poincaré characteristic of M , and in particular is an integer and is zero inthe odd-dimensional boundaryless case.Since a = − b = (2 π ) − , one has L n − ( M ) = − π Z M scal vol M + 12 π Z ∂M (tr S ) vol ∂M (12)where scal denotes the scalar curvature of M . Note that tr S is ( n −
1) times the meancurvature of ∂M in M .If M is flat, that is, R = 0, then Formula (5) simplifies, since only the summandcorresponding to m = 0 may be nonzero, giving L n − k ( M ) = b k Z ∂M tr (cid:18)^ k − S (cid:19) vol ∂M (13)4or 1 ≤ k ≤ n . In particular, L n − ( M ) = 12 π Z ∂M (tr S ) vol ∂M , and (14) χ ( M ) = b n Z ∂M (det S ) vol ∂M if n ≥ . (15)Note that tr S is ( n −
1) times the mean curvature, and det S “the” Lipschitz–Killing, orGauss–Kronecker, curvature, of ∂M in M . If f : M → R is a function on a set and a ∈ R , then the a -sublevel set of f is defined by M af := f − (cid:0) ] −∞ , a ] (cid:1) (16)also written M a if there is no risk of confusion, and the a -level set of f is f − ( a ).Let M be a manifold, let p ∈ N ≥ , and let f ∈ C p ( M, R ). The real number a ∈ R is a regular value of f if f ( x ) = a implies d f ( x ) = 0 for all x ∈ M . We define the setsReg p ( M, R ) := { ( f, a ) ∈ C p ( M, R ) × R | a is a regular value of f } , (17)Reg pc ( M, R ) := { ( f, a ) ∈ Reg p ( M, R ) | M af is compact } . (18)We also set C pa − reg ( M, R ) := pr (cid:0) Reg p ( M, R ) ∩ ( C p ( M, R ) ×{ a } ) (cid:1) and similarly for C pa − reg ,c ( M, R ). Proposition 2.1.
Let M be a manifold, let p ∈ N ≥ , and let ( f, a ) ∈ Reg p ( M, R ) . Then, M af is a full-dimensional C p -submanifold with boundary of M . Its manifold boundary,equal to its topological boundary, is the C p -hypersurface ∂M af = f − ( a ) .Proof. By the submersion theorem, f − ( a ) is a C p -hypersurface of M . By consideringseparately points x ∈ M such that f ( x ) < a and such that f ( x ) = a , one checks that M af is a full-dimensional C p -submanifold with boundary of M , and that ∂M af = f − ( a ). Let (
M, g ) be an n -dimensional Riemannian manifold (not necessarily compact, but with-out boundary). Let f ∈ C ( M, R ). Its gradient is defined by ∇ f := (d f ) ] . Its Hessian is defined by Hess( f ) := ∇ lc d f = ( ∇ lc ∇ f ) [ . Its Laplacian is the trace of its Hessian,∆ f := tr(Hess( f )).Let ( f, a ) ∈ Reg ( M, R ). By Proposition 2.1, the set M a is a full-dimensional C -submanifold with boundary of M and its boundary is the C -hypersurface ∂M a . Theoutward unit normal vectorfield of ∂M a is ν = ∇ f k∇ f k . Therefore, one has ∇ lc ν = k∇ f k − ∇ lc ∇ f + d k∇ f k − (cid:12) ∇ f , which has tangential component k∇ f k − ∇ lc ∇ f .Therefore, the second fundamental form of ∂M a in M is given by S = Hess( f ) | ∇ f ⊥ k∇ f k . (19)We briefly explain the idea underlying the rest of this subsection. By Formula (19),the integrals in the sum on m in Formula (5) are equal to Z ∂M a k∇ f k m +1 − k tr (cid:18)^ m R | ∇ f ⊥ ∧ ^ k − − m Hess( f ) | ∇ f ⊥ (cid:19) vol ∂M a . (20)5e will convert these integrals on ∂M a into integrals on M a by using the divergencetheorem. To do this, we need to find a vectorfield X ∈ X ( M a ) such that X | ∂M a = ν = ∇ f k∇ f k and k∇ f k m +1 − k tr (cid:16)V m R | ∇ f ⊥ ∧ V k − − m Hess( f ) | ∇ f ⊥ (cid:17) X has a divergence whichis integrable on M a . Besides the boundary, the possibly problematic points are the pointswhere ∇ f = 0, first because of the factor k∇ f k m +1 − k , and also because of the restrictionto ∇ f ⊥ . Since the two regions of interest are at f = a and at ∇ f = 0, it makes senseto look for a vectorfield of the form X = F ◦ ∇ fa − f where F ∈ C ( T M, T M ) is such that F ( u ) ∼ ∞ u k u k and F vanishes sufficiently fast at 0 for the divergence to be integrable.We now make this idea precise. Lemma 2.2.
For all n, k, m ∈ N such that ≤ k ≤ n and ≤ m ≤ b k − c , thereexists a homogeneous polynomial P n,k,m with integer coefficients such that for any n -dimensional Euclidean space with orthonormal basis ( V , B ) , any symmetric bilinear forms R ∈ J V V ∗ and H ∈ J V ∗ , and any v ∈ V \ { } , one has tr (cid:18)^ m R | v ⊥ ∧ ^ k − − m H | v ⊥ (cid:19) = P n,k,m (cid:0) ( r ijkl ) , ( h ij ) , ( v i ) (cid:1) k v k k − (21) where ( r ijkl ) (resp. ( h ij ) and ( v i ) ) are the coefficients of R (resp. H and v ) in B . For the sake of definiteness, if B = ( e i ) ≤ i ≤ n is an orthonormal basis of V , we considerthe basis ( e i ∧ e j ) ≤ i Let n, k, m, ( V , B ) , R, H, v be as in the statement. Without loss of generality, wecan suppose that ( V , B ) = ( R n , std) with the standard inner product. Set a i := qP ij =1 v i for 1 ≤ i ≤ n . In particular, a = v and a n = k v k . We first assume that v > 0. Let P bethe following change of basis matrix: P i := v i / k v k , and if j ≥ 2, then P ij := β ij / ( a j − a j )with β ij := − v i v j if i < j , or a j − if i = j , or 0 if j < i . It is an orthogonal matrixand P − HP restricted to the rows and columns 2 ≤ i, j ≤ n is the matrix of H | v ⊥ in anorthonormal basis.Since the β ij ’s are polynomials in the v k ’s and a k ’s, the coefficients ( P − HP ) ij with2 ≤ i, j ≤ n are of the form ( P − HP ) ij = (cid:0) polynomial (cid:1) a i − a i a j − a j . The change of basis matrix in V V associated with P , say Q , has coefficients Q ijkl = P ik P jl − P il P jk . As with P , the matrix Q − RQ restricted to the rows and columns 2 ≤ i, j, k, l ≤ n (with i < j and k < l ) is the matrix of R | v ⊥ in an orthonormal basis. Thecoefficients ( Q − RQ ) ijkl with 2 ≤ i, j, k, l ≤ n (with i < j and k < l ) are of the form( Q − RQ ) ijkl = (cid:0) polynomial (cid:1) a i − a i a j − a j a k − a k a l − a l . The coefficients of the exterior product of their exterior powers is again of a similarform, hence so is its trace. More precisely, it is a rational fraction with variables r ijkl , h ij , v i , a i . The denominator is a product of a i ’s, where the exponent of a n is at most2(2 m + ( k − − m )) = 2( k − v > 0, but it is intrinsic to( R, S, v ) and invariant under orthogonal transformations of V . Therefore, it also holds if v = e n , in which case all the a i ’s with i < n vanish. As a consequence, the only a i ’s atthe denominator are those with i = n , that is, a n = k v k .6he variable a n does not appear in the numerator (since β ij only involves a j − ). For i < n , then a i as a function of the v k ’s is not differentiable at e n but is differentiable at e ,so by invariance under orthogonal transformation, the variables a i with i < n can onlyappear in the numerator with even exponents. Therefore, the numerator is a polynomialin the coefficients r ijkl , h ij , v i . Definition 2.3. For all n, k, m ∈ N such that 1 ≤ k ≤ n and 0 ≤ m ≤ b k − c , we define P n,k,m to be the (unique) polynomial whose existence is asserted in Lemma 2.2. For othervalues of the indices, we set P n,k,m := 0. We set P n,k := P n,k, . Remark . One has dim J V ∗ = n ( n +1)2 and dim J V V ∗ = n ( n − n ( n − . There-fore, P n,k,m has n + [ k − − m ≥ n ( n +1)2 + [ m ≥ n ( n − n ( n − variables. Byhomogeneity considerations, P n,k,m has degree 2( k − 1) in the coefficients of v , degree k − − m in the coefficients of H , and degree m in the coefficients of R .The proof shows how to compute the P n,k,m ’s. For instance, one has( P − HP ) ij = 1 a i − a i a j − a j × a i − h ij a j − + v i v j i − X k =1 j − X l =1 v k h kl v l − a i − v j j − X l =1 h il v l − a j − v i i − X k =1 v k h kj for 2 ≤ i, j ≤ n , and the trace of an exterior power can be computed as a sum of minorsof given order. We consider a few special cases: • If k = 1 (hence m = 0), then the left-hand side of (21) is the trace of the identityon V v ⊥ ’ R , so P n, = 1. • For the case k = 2 (hence m = 0), note that tr( H | v ⊥ ) = tr H − k v k − H ( v, v ). Thisgives P n, = n X i =1 X j = i v j h ii − X i 0, the left-hand side converges to R M (cid:15) (div X ) vol M by Lebesgue’s dominated conver-gence theorem, since div X ∈ L ( M ). The right-hand side is equal, by change of variable,to R ∂M h ϑ ∗ (cid:15) X, v i (det T ϑ (cid:15) ) vol ∂M , which converges to R ∂M h X, v i vol ∂M since the integrand isuniformly convergent and ∂M is compact.We can now prove a first general result. Theorem 2.6. Let ( M, g ) be an n -dimensional Riemannian manifold. Let ( f, a ) ∈ Reg c ( M, R ) . For ≤ k ≤ n and ≤ m ≤ b k − c , let F k,m ∈ C ( T M, T M ) be suchthat F k,m ( u ) ∼ ∞ u k u k . Then, for ≤ k ≤ n , one has L n − k ( M a ) = a k Z M a tr (cid:18)^ k/ R (cid:19) vol M + b k − c X m =0 b k,m Z M a div (cid:18) k∇ f k m +3(1 − k ) P n,k,m ( R, Hess( f ) , ∇ f ) (cid:18) F k,m ◦ ∇ fa − f (cid:19)(cid:19) vol M (28) under the condition that the divergence appearing in the integral exists and is integrable.If M is flat, then for ≤ k ≤ n , let F k ∈ C ( T M, T M ) be such that F k ( u ) ∼ ∞ u k u k .Then, L n − k ( M a ) = b k Z M a div (cid:18) k∇ f k − k ) P n,k (Hess( f ) , ∇ f ) (cid:18) F k ◦ ∇ fa − f (cid:19)(cid:19) vol M . (29) under the same conditions. emark . By “ F ( u ) ∼ ∞ u k u k ”, we mean that lim k u k→ + ∞ d (cid:16) F ( u ) , u k u k (cid:17) = 0, where d isthe distance on T M induced by the Riemannian metric of M (or any distance, since M a is compact and u k u k has unit norm). Proof. Starting with the definition (5), we use the expression of the second fundamentalform (19) and Equation (23) to obtain L n − k ( M ) := a k Z M tr (cid:18)^ k/ R (cid:19) vol M + b k − c X m =0 b k,m Z ∂M P n,k,m ( R, Hess( f ) , ∇ f ) k∇ f k k − − m vol ∂M . (30)The asymptotic property of F k,m ensures that the vectorfield whose divergence is consid-ered in the statement is continuous on ∂M a and its value there is k∇ f k m +3(1 − k ) P n,k,m ( R, Hess( f ) , ∇ f ) ∇ f k∇ f k . Finally, the hypotheses of the proposition ensure that the divergence theorem applies. Remark . Since P n, = 1, the theorem for k = 1 holds for ( f, a ) ∈ Reg c ( M, R ).Our next step is to find explicit functions F (in particular proving that some exist)making the divergence appearing in the theorem integrable. We consider radial mapsof the form F k,m ( u ) = k u k k − − m G k,m ( k u k ) u with G k,m ∈ C ( R ≥ , R ). The condi-tion G k,m ( x ) ∼ + ∞ x m +1) − k ensures that F k,m ( u ) ∼ ∞ u k u k . Examples of functions G satisfying these conditions are given by G k,m ( x ) := (cid:16) x k − m +1)) (cid:17) − / . We set η f,‘ := q f ‘ + k∇ f k ‘ (31)for ‘ ≥ 0. These choices for F yield the following theorem. Theorem 2.9. Let ( M, g ) be an n -dimensional Riemannian manifold. Let ( f, a ) ∈ Reg c ( M, R ) . For ≤ k ≤ n , one has L n − k ( M a ) = a k Z M a tr (cid:18)^ k/ R (cid:19) vol M + b k − c X m =0 b k,m Z M a div P n,k,m ( R, Hess( f ) , ∇ f ) η f − a, k − m +1) ∇ f ! vol M . (32) If M is flat and ≤ k ≤ n , then L n − k ( M a ) = b k Z M a div P n,k (Hess( f ) , ∇ f ) η f − a, k − ∇ f ! vol M . (33)For k = 1 , 2, this gives L n − ( M a ) = 12 Z M a div ∇ fη f − a ! vol M , (34) L n − ( M a ) = − π Z M a scal vol M +12 π Z M a div k∇ f k ∆ f − Hess( f )( ∇ f, ∇ f ) η f − a, ∇ f ! vol M . (35)9imilarly, when M is flat and Hess( f ) | ∇ f ⊥ is nondegenerate, if n ≥ 1, one has χ ( M a ) = b n Z M a div det(Hess( f )) k∇ f k n Hess( f ) (cid:0) ∇ f, ( ∇ f ) Hess( f ) (cid:1) η f − a, n − ∇ f ! vol M . (36) Remark . There are obviously many natural choices for the functions F and G . Forinstance, one can take F k,m := F k . With the F k ’s given above, the divergence correspond-ing to the m th summand reads div (cid:16) k∇ f k m P n,k,m ( R, Hess( f ) , ∇ f ) η f − a, k − ∇ f (cid:17) . In the case of nodalvolumes, other choices are given in the next subsection. Remark . Intrinsic volumes can be defined for Riemannian manifolds with corners,and even Whitney stratified spaces of “positive reach” in Riemannian manifolds. Sincethe divergence theorem admits generalizations to these settings, it is possible to extendthe above results to sublevel sets of functions defined on Riemannian manifolds withboundary or corners, and to Whitney stratified spaces in Riemannian manifolds, underthe assumption that the function is transverse to the boundary or the strata respectively.Boundary terms will appear in the formulas. We do not carry out this generalization infull and only give a formula for nodal volumes in the next subsection (see Remark 2.13). In this subsection, we show how we can compute the intrinsic volumes of the zero sets,or nodal sets, of functions defined on compact Riemannian manifolds. Let ( M, g ) be acompact n -dimensional Riemannian manifold. Let f ∈ C − reg ( M, R ) (class C is sufficientby Remark 2.8). The zero set of f is Z f := f − (0) = ∂M f = ∂M − f . By (10), onehas vol( Z f ) = L n − ( M f ) + L n − ( M − f ). Since M f ∪ M − f = M and M f ∩ M − f = Z f is negligible in M , Formula (34) gives an integral on M . Using the general formula ofTheorem 2.6 yields vol( Z f ) = − Z M div (cid:18) F ◦ ∇ f | f | (cid:19) vol M (37)(where minus the absolute value appears since f is negative on M f and positive on M − f ).Of course, this identity could have been obtained directly by applying the divergencetheorem to the identity vol( Z f ) = R ∂M f vol ∂M f .Recalling the definition of η f,‘ by Equation (31), we set η f := η f, = q f + k∇ f k . (38)We also write σ f : M → {− , , } for the sign of f .Setting, in Formula (37), F ( u ) := G ( k u k ) u with respectively G ( x ) := (1 + x ) − / and π arctan xx and tanh xx , one obtainsvol( Z f ) = 12 Z M σ f η f (cid:16) f k∇ f k + Hess( f )( ∇ f, ∇ f ) − η f ∆ f (cid:17) vol M (39)andvol( Z f ) = 1 π Z M k∇ f k − (cid:18) arctan ◦ k∇ f k f (cid:19) Hess( f )( ∇ f, ∇ f ) k∇ f k − ∆ f ! + η − f k∇ f k − f Hess( f )( ∇ f, ∇ f ) k∇ f k !! vol M (40)10ndvol( Z f ) = 12 Z M k∇ f k − (cid:18) tanh ◦ k∇ f k f (cid:19) Hess( f )( ∇ f, ∇ f ) k∇ f k − ∆ f ! + (cid:18) cosh ◦ k∇ f k f (cid:19) − k∇ f k f − Hess( f )( ∇ f, ∇ f ) f k∇ f k !! vol M (41)(see [15] for the computation details). Remark . In the last three formulas, all terms of the integrands are bounded on M andcontinuous on M \ Z f . Indeed, the Hessian expressions are quadratic in k∇ f k , the arctanand tanh expressions are linear in k∇ f k when k∇ f k is small, and the cosh expression isexponentially small in | f | when | f | is small. However, not all terms need be continuouson M . This problem is dealt with below. Remark . Fulfilling the promise made in Remark 2.11, let M be a compact Riemannianmanifold with boundary. If f intersects ∂M transversely, then Formula (37) becomesvol( Z f ) = 12 (cid:18)Z ∂M (cid:28) F ◦ ∇ ff , v (cid:29) vol ∂M − Z M (cid:18) div (cid:18) F ◦ ∇ ff (cid:19)(cid:19) vol M (cid:19) . (42)Note that by the transversality assumption, Z f ∩ ∂M is negligible in ∂M . This formulareduces in dimension 1 to [3, Prop. 3]. Remark . The cases considered in [3] correspond to M = ( R / Z ) n with the standardflat metric. In particular, Formula (39) is essentially [3, Prop. 5] (in the case of ( R / Z ) n with the standard flat metric). In dimension 1, the general formula (37) reduces to [3,Prop. 2], and in that case, only the condition lim x →±∞ F ( x ) = ± L n − k ( A ) + L n − k ( B ) = L n − k ( A ∪ B ) − L n − k ( A ∩ B ) (43)when A, B are subsets of a compact n -dimensional Riemannian manifold M such that allterms are well-defined (see [10, Thm. 5.16(6)]). Therefore, if f ∈ C − reg ( M, R ), then L n − k ( Z f ) = L n − k ( M f ) + L n − k ( M − f ) − L n − k ( M ) . (44)The terms corresponding to the first summand in (32) cancel out, so that L n − k ( Z f ) = b k − c X m =0 b k,m Z M σ kf div P n,k,m ( R, Hess( f ) , ∇ f ) η f, k − m +1) ∇ f ! vol M . (45)The exponent k of σ f is congruent modulo 2 to deg Hess( f ) P n,k,m + deg ∇ f P n,k,m + 1 =3 k − m ) by Remark 2.4. In particular, L n − k ( Z f ) = 0 for k even, as expected. Onecan also consider Z f as a Riemmannian manifold with curvature ˜ R and obtain L n − − k ( Z f ) = a k Z Z f tr (cid:18)^ k/ ˜ R (cid:19) vol Z f (46)where ˜ R is given by the Gauss formula for the curvature of submanifolds, ˜ R ( X, Y, Z, T ) = R ( X, Y, Z, T ) + S ( X, Z ) S ( Y, T ) − S ( X, T ) S ( Y, Z ) for X, Y, Z, T ∈ X ( Z f ).11e return to the question raised in Remark 2.12 of having continuous integrands. Theonly non-continuous terms in the integrands of Equations (39), (40), (41) are of the form σ f h (cid:16) Hess( f )( ∇ f, ∇ f ) − ∆ f k∇ f k (cid:17) (47)with h ∈ C ( M, R ), respectively h = η − f and h = k∇ f k − (cid:16) arctan ◦ k∇ f k| f | (cid:17) and h = k∇ f k − (cid:16) tanh ◦ k∇ f k| f | (cid:17) . This is dealt with in [3] (in the case of Equation (39) on a flattorus) using an integration by parts. The same method extends to compact Riemannianmanifolds as follows. One hasHess( f )( ∇ f, ∇ f ) − (∆ f ) k∇ f k = h∇ f, ∇ lc ∇ f ∇ f − (∆ f ) ∇ f i . Therefore, σ f h (cid:16) Hess( f )( ∇ f, ∇ f ) − (∆ f ) k∇ f k (cid:17) = (cid:10) ∇ | f | , h (cid:0) ∇ lc ∇ f ∇ f − (∆ f ) ∇ f (cid:1)(cid:11) . We temporarily assume that f is of class C and we use the fact that div ( | f | h ( ∇ lc ∇ f ∇ f − (∆ f ) ∇ f ))has a vanishing integral on M (by the standard divergence theorem). Therefore, Z M σ f h (cid:16) Hess( f )( ∇ f, ∇ f ) − (∆ f ) k∇ f k (cid:17) vol M = Z M | f | div ( h ((∆ f ) ∇ f − ∇ lc ∇ f ∇ f )) vol M . One has div((∆ f ) ∇ f ) = (∆ f ) + h∇ ∆ f, ∇ f i . The Bochner formula yieldsdiv ( ∇ lc ∇ f ∇ f ) = div (cid:18) ∇k∇ f k (cid:19) = 12 ∆ k∇ f k = h∇ ∆ f, ∇ f i + k Hess f k + Ric( ∇ f, ∇ f )where the norm of the Hessian is the Hilbert–Schmidt norm. Therefore, the third deriva-tives cancel out. Since C ( M, R ) is dense in C ( M, R ) for the (Whitney) strong C -topology(see for instance [14, Thm. II.2.6], and the next section for function space topologies) andthe involved quantities are continuous in this topology, one has, for any f of class C , Z M σ f h (cid:16) Hess( f )( ∇ f, ∇ f ) − (∆ f ) k∇ f k (cid:17) vol M = Z M | f | (cid:16) h (cid:16) (∆ f ) − k Hess f k − Ric( ∇ f, ∇ f ) (cid:17) + h∇ h, (∆ f ) ∇ f − ∇ lc ∇ f ∇ f i (cid:17) vol M . (48)For example, one has ∇ η − f = − η − f ( f ∇ f + ∇ lc ∇ f ∇ f ), so Formula (39) becomesvol( Z f ) = 12 Z M | f | η f (cid:16) k∇ f k − | f | ∆ f + (∆ f ) − k Hess f k − Ric( ∇ f, ∇ f ) (cid:17) +3 η − f (cid:16) f Hess( f )( ∇ f, ∇ f ) + Hess( f )( ∇ f, ∇ lc ∇ f ∇ f ) − (∆ f ) ( f k∇ f k + Hess( f )( ∇ f, ∇ f )) (cid:17) ! vol M . (49)In the case of a flat torus, this is [3, Prop. 7].12 emark . These formulas can also be written in terms of the tracefree Hessian. Recallthat Hess ( f ) = Hess( f ) − ∆ fn id. A tracefree linear map is Hilbert–Schmidt-orthogonalto the identity, so k Hess f k = (∆ f ) n + k Hess f k .In Equation (49), the integrand is a Lipschitz continuous functional of f ∈ C − reg ( M, R )(see next section for the precise setting), so one can apply techniques of the Malliavincalculus (see [3]). The only difference between (49) and [3, Prop. 7] is the additional terminvolving the Ricci curvature, | f | η − f Ric( ∇ f, ∇ f ), and this term is in the required domainof the Malliavin calculus by the same proof as [3, Lem. 2 p. 26]. Therefore, [3, Thm. 1]holds on any compact Riemannian manifold. Similarly, Formula (42) shows that the extraboundary terms are not problematic, so [3, Thm. 1] holds on any compact Riemannianmanifold with corners, a generalization which includes [3, Thm. 2] as a special case. For this subsection, we refer to [14, Ch. II] for details. Let ( M, g ) be a Riemannianmanifold and p ∈ N . We will use two different topologies on the set C p ( M, R ), the uniform C p -topology , and the finer (Whitney) strong C p -topology . The resulting topologicalspaces will be denoted with the subscripts U and S respectively. The first is a completelymetrizable group and the second is a Baire topological group (countable intersections ofdense open subsets are dense). In particular, there is a notion of Lipschitz continuity(by which we mean “local Lipschitz continuity” ) for maps between the first space andother metric spaces. The product C p ( M, R ) × R will be considered with the correspondingproduct topology, and the sets Reg p ( M, R ) and Reg pc ( M, R ) defined in Equations (17)and (18) with the corresponding subspace topologies.If 0 ≤ i ≤ p and f ∈ C p ( M, R ), then ∇ lc i f ∈ Γ ( p − i ) ( N i T M → M ). We denoteby k∇ lc i f ( x ) k the norm of this multilinear form induced by the norm g x on T ∗ x M , andby k∇ lc i f k ∞ the supremum of these norms for x ∈ M . Let p ∈ N . For the uniform C p -topology, a neighborhood basis of 0 is given by U p ( (cid:15) ) := { f ∈ C p ( M, R ) | p X i =0 k∇ lc i f k ∞ < (cid:15) } (50)for (cid:15) ∈ R > . For the strong C p -topology, a neighborhood basis of 0 is given by S p ( (cid:15) ) := { f ∈ C p ( M, R ) | ∀ x ∈ M p X i =0 k∇ lc i f ( x ) k < (cid:15) ( x ) } (51)for (cid:15) ∈ C ( M, R > ). The uniform and strong C ∞ -topologies are obtained as the unions ofthe corresponding C p -topologies.The strong C p -topology does not depend on the Riemannian metric (it could actually bedefined using norms of usual derivatives in charts). These topologies differ in the control offunctions at infinity (in particular, they are equal when M is compact). Results involvingthe strong topology will often remain true for the uniform topology when restricted toproper functions.The strong topology has the disadvantage that the inclusion of constant functions, R → C p ( M, R ) S , a ( x a ), is not continuous. For example, the function τ : C p ( M, R ) U × R −→ C p ( M, R ) U ( f, a ) f − a. (52) Note that (local) Lipschitz continuity does not imply uniform continuity when the domain is notcomplete, as the sign function on R =0 shows. 13s Lipschitz continuous, but the analogous result (for mere continuity) with the strongtopology does not hold. Therefore, when studying sublevel sets at varying heights, we willuse the uniform topology and we will restrict our attention to proper functions, and whenstudying sublevel sets at a fixed height, we will use the strong topology if we want to allownonproper functions.We denote by C p p ( M, R ) (resp. C p b ( M, R )) the set of proper (resp. bounded below)functions in C p ( M, R ), and by C p pb ( M, R ) := C p p ( M, R ) ∩C p b ( M, R ) the set of proper boundedbelow functions. Similarly, we set Reg p ∗ := Reg p ( M, R ) ∩ ( C p ∗ ( M, R ) × R ) for ∗ = b , p , pb. Proposition 3.1. Let p ∈ N . At least one (resp. all) sublevel set(s) of f ∈ C p ( M, R ) is/arecompact if and only if f is bounded below (resp. proper bounded below). In particular, Reg p pb ( M, R ) ⊆ Reg pc ( M, R ) . The subsets C p b ( M, R ) and C p p ( M, R ) and C p pb ( M, R ) are openand closed in C p ( M, R ) U .Proof. Obvious. Example . For a given function, the set of real numbers such that the associated sublevelset is compact can be any downset. Indeed, consider the functions on R which send x torespectively x or a or a + e x or x . The sets of real numbers such that the associatedsublevel set is compact are ∅ and ] −∞ , a [ and ] −∞ , a ] and R respectively. If in the secondcase one requires that a be a regular value, then consider x a − e x .Recall that η f was defined by Equation (38) and τ by Equation (52). Lemma 3.3. The function inf : C ( M, R ) U → R ∪ {−∞} is Lipschitz continuous. Thefunction η : C ( M, R ) U → C ( M, R ) U is Lipschitz continuous. Let p ∈ N . The function η : C p − reg ( M, R ) X → C p ( M, R > ) X is Lipschitz continuous for X = U and continuous for X = S . The function m := inf ◦ η ◦ τ : C p ( M, R ) U × R → R is Lipschitz continuous.Proof. Obvious. Proposition 3.4. Let p ∈ N ≥ . The subset Reg p p ( M, R ) is open and dense in C p p ( M, R ) U × R . The subset C p − reg ( M, R ) (resp. C p − reg ,c ( M, R ) ) is open and dense in C p ( M, R ) S (resp. C p − c ( M, R ) S ). The three openness results actually hold for the (uniform or strong) C -topology.Proof. One has, Reg p p ( M, R ) = (cid:16) m | C p p ( M, R ) U (cid:17) − ( R > ), which is therefore open in C p p ( M, R ) U .Similarly, C p − reg ( M, R ) = η − (cid:0) C p − ( M, R > ) (cid:1) is open in C p p ( M, R ) S .As for density, by the Morse–Sard theorem, if f ∈ C n ( M, R ), then the set of regularvalues of f is dense. This implies that Reg p p ( M, R ) is dense in C max( n,p ) ( M, R ) U × R , whichis dense in C p ( M, R ) U × R (see [14, Thm. II.2.6]).For C p − reg ( M, R ), one can use the transversality theorem as follows. Let f ∈ C p ( M, R )and (cid:15) ∈ C ( M, R > ). Let ( φ i ) i ∈ I be a smooth partition of unity subordinated to somelocally finite atlas of M . Consider the map Φ : M × R | I | → R , ( x, ( λ i )) f ( x ) + (cid:15) ( x ) P i λ i φ i ( x ). Then, Φ is submersive, so for almost all tuples ( λ i ), the map Φ( − , ( λ i ))is transverse to 0.The proofs work similarly with the compactness requirement added. We begin with the special cases of the volume and the nodal volume, which will be neededin the proof of the general case. We actually prove a more general statement, where denotes the symmetric difference of two sets.14 roposition 3.5. Let ( M, g ) be a Riemannian manifold. The functions Reg ( M, R ) U −→ R ≥ (53) (cid:0) ( f, a ) , ( g, b ) (cid:1) vol( M af M bg ) and Reg ( M, R ) U −→ R ≥ (54)( f, a ) vol( M af ) are continuous (resp. Lipschitz continuous) when the domains are given the uniform C (resp. C )-topology. The functions C − reg ,c ( M, R ) S −→ R ≥ (55)( f, g ) vol( M f M g ) and C − reg ,c ( M, R ) S −→ R ≥ (56) f vol( M f ) are continuous when the domains are given the strong C -topology.The function Reg ( M, R ) U −→ R ≥ (57)( f, a ) vol( ∂M af ) is continuous when the domain is given the uniform C -topology.Remark . The continuity of nodal volumes was proved in the Euclidean case in [4,Thm. 3], with a similar proof.We first prove a lemma. Lemma 3.7. Let f ∈ C − reg ,c ( M, R ) . For any neighborhood U of Z f , there exists anopen neighborhood V of f in the strong C -topology (and, if f is proper, in the uniform C -topology) such that for any h, k ∈ V , one has Z h ⊆ U and M h M k ⊆ U .Proof. Let f and U be as in the statement. Then M f is compact. Let H, K be compactsubsets such that M f \ U ⊆ H ⊂⊂ M f ⊂⊂ K ⊆ M f ∪ U . By Proposition 2.1, f is strictlynegative on int M f , so (cid:15) := inf {− f ( x ) | x ∈ H } > (cid:15) := inf { f ( x ) | x ∈ M \ K } > V := f + U (min( (cid:15) , (cid:15) )). In the nonproper case, let (cid:15) := min( (cid:15) , inf { f ( x ) | x ∈ ∂K } ) > (cid:15) ∈ C ( M, R > ) be the function equal to (cid:15) on K and min( (cid:15) , f ) on M \ K , and set V := f + S ( (cid:15) ). In both cases, if h ∈ V , one has Z h ⊆ K \ H ⊆ U , and similarly M h M k ⊆ U if h, k ∈ V . Proof of the proposition. (i) Let Φ be the first function in the proposition. Since M af = M τ ( f,a ) and τ is Lipschitz continuous, it suffices to consider 0-sublevel sets. Let (( f, , ( g, ∈ Reg ( M, R ) . If (( h, , ( k, ∈ Reg ( M, R ) , then ( M f M g ) ( M h M k ) ⊆ ( M f M h ) ∪ ( M g M k ), so | Φ(( f, , ( g, − Φ(( h, , ( k, | ≤ Φ(( f, , ( h, g, , ( k, − , (0 , − − , τ , we canrestrict our attention to ( f, ∈ Reg ( M, R ) U . Then, Z f is a compact C -hypersurface.For any x ∈ Z f , there exists a smooth chart φ : U → R n such that φ ( U ) = ]0 , d and φ ( Z f ∩ U ) is the graph of a C -function f φ : ]0 , d − → ]1 / , / 3[ with k d f φ k ∞ < 1. Since Z f is compact, there exists a finite cover ( U i ) i ∈ I of Z f by such sets. Let ( ψ i ) i ∈ I be asmooth partition of unity subordinated to ( U i ) i ∈ I .By Lemma 3.7, there exists a neighborhood V of f such that the nodal set of any h ∈ V is included in S i ∈ I U i . For any i ∈ I , by the implicit function theorem, there existsa neighborhood V i ⊆ V of f in the C -topology such that for any h ∈ V i , the hypersurface φ i ( U i ∩ Z h ) is the graph of a function h i : ]0 , d − → ]0 , 1[ with k d h i k ∞ < 2. One thenhas ∇ h i = − (cid:16) ∂ ( φ i ∗ h ) ∂x d (cid:17) − ∇ ( φ i ∗ h ). Set W := T i ∈ I V i .Let h ∈ W and let j : Z g , → M be the inclusion. Then,vol( Z h ) = X i ∈ I Z U i ∩Z h j ∗ ( ψ i vol M )= X i ∈ I Z ]0 , d − q k∇ h k φ i ∗ j ∗ ( ψ i vol M )= X i ∈ I Z ]0 , d − s (cid:18) ∂ ( φ i ∗ h ) ∂x d (cid:19) − k∇ ( φ i ∗ h ) k (pr d − ◦ φ i ) ∗ ( ψ i vol M )where pr d − : R d → R d − is the projection on the first d − Z − ) for the uniform C -topology is clear.(v) We now prove the Lipschitz continuity statement. Let ( f, ∈ Reg ( M, R ). ByLemma 3.7, there exists an open neighborhood V of f in the uniform C -topology suchthat Let K be a compact neighborhood of Z f such that (cid:15) := min (cid:0) inf {| f ( x ) | | x ∈ M \ K } , inf {k∇ f ( x ) k | x ∈ K } (cid:1) > 0. Let h, k ∈ C − reg ,c ( M, R ) ∩ ( f + U ( (cid:15) )). Then, M h M k ⊆ K . Furthermore, M h M k is included in a tubular neighborhood of Z h ofradius (cid:15) k h − k k ∞ . Indeed, let x ∈ M h M k . One can suppose that k ( x ) ≤ ≤ h ( x ).Therefore, 0 ≤ h ( x ) ≤ k h − k k ∞ . Consider the maximal integral curve c defined by c (0) = x and c ( t ) = − ∇ h ( c ( t )). Let T := inf { t ≥ | c ( t ) ∈ Z h } . One has c ([0 , T ]) ⊆ K .Since k∇ h ( c ( t ) k ≥ k∇ f ( c ( t )) k − (cid:15) ≥ (cid:15) , one has d ( c (0) , c ( T )) ≤ (cid:15) k h − k k ∞ .Since by the previous part of the proof, the nodal volume is continuous, vol( Z h ) isbounded, say by A > 0, on a C -neighborhood W of f . Therefore, if h, k ∈ C − reg ,c ( M, R ) ∩ ( f + U ( (cid:15) )) ∩ W , then vol( M h M k ) ≤ A (cid:15) k h − k k ∞ where A > A and( M, g ). Remark . The volume function is not uniformly continuous, as the pairs of constantfunctions equal to ± n on a nonempty compact manifold M show: the volume of the 0-sublevel set jumps from 0 to vol( M ) for two arbitrarily close functions. Similar examplescan be given for any nonzero intrinsic volume. Remark . The proof shows that the first two functions are actually pointwise Lipschitzcontinuous when the domain is given the uniform C -topology. Theorem 3.10. Let ( M, g ) be an n -dimensional Riemannian manifold. If ≤ k ≤ n ,then the function L sub n − k : Reg ( M, R ) U −→ R (58)( f, a ) n − k ( M af )16 s Lipschitz continuous, and the function L n − k ( M − ) : C − reg ,c ( M, R ) S −→ R (59) f n − k ( M f ) is continuous.Remark . By Remark 2.8, the functions L sub n − and L n − ( M − ) are also defined for C -functions, and the proof below applies. For mere continuity in the proper case, the resulteven holds for C -functions, as proved in Proposition 3.5. Remark . Since the Euler–Poincaré characteristic is an integer, one obtains that χ ( M −− ) is locally constant on Reg ( M, R ) U and χ ( M − ) is locally constant on C − reg ,c ( M, R ) S . Lemma 3.13. Let ( M, g ) be an n -dimensional Riemannian manifold and p ∈ N . Let n, k, m ∈ N . The function Y n,k,m : Reg p +3 ( M, R ) U −→ C p ( M, R ) U ( f, a ) div P n,k,m ( R, Hess( f ) , ∇ f ) η f − a, k − m +1) ∇ f ! (60) is Lipschitz continuous. The function Y n,k,m ( − , 0) : C p +30 − reg ( M, R ) S → C p ( M, R ) S is con-tinuous.Proof. The result follows from the continuity of the four functions τ : Reg p +3 ( M, R ) U −→ C p +30 − reg ( M, R ) U ,η − , k − m +1) : C p +30 − reg ( M, R ) X −→ C p +2 ( M, R > ) X ,P n,k,m ( R, Hess( − ) , ∇ − ) ∇ − : C p +3 ( M, R ) X −→ X p +1 ( M ) X , div : X p +1 ( M ) X −→ C p ( M, R ) X for X = U, S , with Lipschitz continuity when X = U . Proof of the theorem. Because of the general identity M bf = M af + a − b , it is sufficient toconsider ( f, , ( g, ∈ Reg ( M, R ). By Formula (32), one has (cid:12)(cid:12)(cid:12) L n − k ( M g ) − L n − k ( M f ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M g tr (cid:18)^ k/ R (cid:19) vol M − Z M f tr (cid:18)^ k/ R (cid:19) vol M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + b k − c X m =0 | b k,m | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M g Y n,k,m ( g, M − Z M f Y n,k,m ( f, M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (61)As for the first term, the integrand tr (cid:16)V k/ R (cid:17) is locally bounded on M , and vol( M g M f )is controlled by d ( f, g ) by Proposition 3.5.As for the summands, it suffices to bound R M g M f max( | Y n,k,m ( f, | , | Y n,k,m ( g, | )vol M and R M f ∩ M g | Y n,k,m ( g, − Y n,k,m ( f, | vol M . The first term is dealt with by Proposi-tion 3.5 and the second by the lemma. References [1] Robert J. Adler and Jonathan E. Taylor, Random fields and geometry , Springer Mono-graphs in Mathematics, 2007. 172] Jean-Marc Azaïs and Mario Wschebor, Level sets and extrema of random processesand fields , Wiley, 2009.[3] Jürgen Angst and Guillaume Poly, On the absolute continuity of random nodal vol-umes , arXiv:1811.04795 [math.PR].[4] Jürgen Angst, Guillaume Poly and Hung Pham Viet, Universality of the nodal lengthof bivariate random trigonometric polynomials , arXiv:1610.05360 [math.PR].[5] Andreas Bernig and Ludwig Bröcker, Lipschitz–Killing invariants , Math. Nachr. (2002), 5–25.[6] S.-S. Chern, A simple intrinsic proof of the Gauss–Bonnet formula for closed Rie-mannian manifolds , Ann. Math. (1944), 747–752.[7] S.-S. Chern, On the curvatura integra in a Riemannian manifold , Ann. Math. (4)(1945), 674–584.[8] S.-S. Chern, On the kinematic formula in the Euclidean space of n dimensions , Amer.J. Math. (1952), 227–236.[9] S.-S. Chern, On the kinematic formula in integral geometry , J. Math. and Mech. (1966), 101–118.[10] H. Federer, Curvature Measures , Trans. Amer. Math. Soc. (1959), 418–491.[11] Joseph H. G. Fu, Curvature measures and generalized Morse theory , J. Diff. Geom. (1989), 619–642.[12] Joseph H. G. Fu, Curvature measures and Chern classes of singular varieties , J. Diff.Geom. (1994), 251–280.[13] Joseph H. G. Fu and Thomas Wannerer, Riemannian curvature measures , Geom.Funct. Anal., in press, arXiv:1711.02155 [math.DG].[14] Morris W. Hirsch, Differential Topology , Graduate Texts in Mathematics, Vol. 5,Springer-Verlag, 1997.[15] Benoît Jubin, Closed Kac–Rice type formulas on Riemannian manifolds ,arXiv:1901.01629 [math.DG].[16] Raphaël Lachièze-Rey, Two-dimensional Kac-Rice formula. Application to shot noiseprocesses excursions , arXiv:1607.05467 [math.PR].[17] Jean-Marie Morvan, Generalized Curvatures , Geometry and Computing, Springer,2008.[18] Liviu I. Nicolaescu, On the Kac–Rice formula , available at .[19] Michael Spivak, A Comprehensive Introduction to Differential Geometry, Volume 5 ,Publish or Perish, 1999.[20] Christoph Thäle, Fifty years of sets with positive reach, a survey , Surveys in Mathe-matics and its Applications (2008), 123–165.[21] Hermann Weyl, On the volume of tubes , Amer. J. Math., (1939), 461–472.1822] Martina Zähle, Curvature Measures and Random Sets, I , Math. Nachr. (1984),327–339.[23] Martina Zähle,