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FRAME BUNDLE APPROACH TO GENERALIZED MINIMAL SUBMANIFOLDS KAMIL NIEDZIAL OMSKI Abstract. We extend the notion of r–minimality of a submanifold in arbi- 6 1 trary codimension to u–minimality for a multi–index u ∈ Nq, where q is the 0 codimension. This approach is based on the analysis on the frame bundle of 2 orthonormal frames of the normal bundle to a submanifold and vector bun- n dles associated with this bundle. The notion of u–minimality comes from the a variation of σ –symmetric function obtained from the family of shape opera- u J tors corresponding to all possible bases of the normal bundle. We obtain the 9 variation ﬁeld, which gives alternative deﬁnition of u–minimality. Finally, we 1 givesomeexamplesofu–minimalsubmanifoldsforsomechoicesofuandstate some relations between generalized symmetric functions σ . ] u G D . h t a m 1. Introduction [ The notion of minimality of a submanifold has long history. It has been con- 2 sidered by many authors within many contexts. Minimality can be deﬁned by v vanishing of the mean curvature – the trace of the second fundamental form. The 8 4 other possible extrinsic conditions we may impose on a submanifold are: total 2 geodesicity, when second fundamental form vanishes, and total umbilicity, when 2 the second fundamental form is proportional to the mean curvature vector. 0 . Whilestudying hypersurfaces, i.e. codimension one submanifolds, we may con- 1 0 sider more invariants, which come from eigenvalues of the second fundamental 6 form. We deﬁne the r–th curvature as a r–th symmetric function of the principal 1 curvatures. This has been extensively considered since Reilly’s work [17]. Let us : v briethly recall his approach. Let L be a hypersurface in a Riemannian manifold i X (M,g) with the Levi–Civita connection ∇. Denote by B the second fundamental r form and by A = AN the shape operator corresponding to a choice of normal a vector N, B(X,Y) = (∇ Y) , g(A(X),Y) = g(B(X,Y),N), X,Y ∈ TL. X ⊥ The symmetric function S of A is deﬁned by the characteristic polynomial r n χ (t) = det(I +tA) = S tr, A r r=0 X 2010 Mathematics Subject Classiﬁcation. 53C40; 53C42. Key words and phrases. Submanifold; mean curvatures; ﬁrst variation; u–minimal subman- ifold; generalized Newton transformation. 1 2 K.NIEDZIAL OMSKI where n = dimL. It is a remarkable observation that the variation of S is r described by the Newton transformation T = T (A). Namely, let r r r T = (−1)jS Ar j, r = 0,1,...,n. r j − j=0 X Then, we have d d (1.1) S (t) = tr A(t)·T , r r 1 dt dt − (cid:18) (cid:19) where S (t) is a one parameter family of symmetric functions corresponding to a r family of operators A(t), which satisﬁes A(0) = A [17]. With the use of the above formula Reilly obtained the variation of the integral of S in the case of M being r of constant sectional curvature. The Euler–Lagrange equation deﬁnes the notion of r–minimality. Studies on submanifolds satisfying certain conditions involving r–th mean curvature has been recently very fruitful [1, 2, 7, 8, 13]. Analogous considerations has been also led in the case of foliations [7, 4]. There has been several attempts to generalize this approach to submanifolds of arbitrary codimension (see [18, 10, 11]) and to the case of arbitrary foliations [9, 5]. The problems which appear are the following. First of all, the normal bundle to a submanifold is not trivial in a sense that the covariant derivative ∇ ⊥ does not annihilate unit vector ﬁelds. Secondly, there is no canonical choice of the orthonormal basis in the normal bundle. The second problem has been overcome (see [18, 14]) by introducing transformations which depend only on the points on the manifold, 1 (T )i = δi1...,irig(B ,B )...g(B ,B ) r j r! j1...,jrj i1j1 i2j2 ir−1jr−1 irjr i1,...,iXr;j1,...,jr for r even, and (Tα)i = δi1...,irig(B ,B )...g(B ,B )(Aα)i r j j1...,jrj i1j1 i2j2 ir−2jr−2 ir−1jr−1 j i1,...,iXr;j1,...,jr for r odd, where δi1,...,ir is the generalized Kronecker symbol and B = B(e ,e ) j1,...,jr ij i j for some choice of orthonormal basis (e ). Then i 1 S = (Tα )i(Aα)j for r even, r r r 1 j i − i,j,α X 1 S = (T )i(Aα)je for r odd, r r r−1 j i α i,j,α X denote the r–th symmetric functions of curvatures (for r odd the r–th mean cur- vature is a vector ﬁeld). In [10] the authors studied r–th minimality and stability with respect to these generalized transformations. The stability conditions led to non–existence results of stable minimal submanifolds on spheres [22, 10]. In this article, we deﬁne the notion of minimality associated with the general- ized Newton transformation introduced by K. Andrzejewski, W. Koz lowski and FRAME BUNDLE APPROACH TO GENERALIZED MINIMAL SUBMANIFOLDS 3 the author [3]. The idea of comes from then deﬁnition of symmetric functions associated with a system A = (A ,...,A ) of matrices (endomorphisms), 1 q χA(t) = σutu, u X where u = (u ,...,u ) ∈ Nq is a multi–index, t = (t ,...,t ) and tu = tu1...,tuq. 1 q 1 q 1 q Generalized Newton transformation T = T (A) depends on the multi–index u u u and a system of endomorphisms A and its recursive deﬁnition is the following T0 = I, Tu = AαTα♭(u), α X where α (u) is a multi–index obtained from u by subtracting 1 in the α–th coor- ♭ dinate. Moreover, it satisﬁes analogue of the condition (1.1), namely, d d σ (t) = tr A (t)·T . dt u dt α α♭(u) α (cid:18) (cid:19) X With the use of above characterization of T we study the variation of σ . u u In this setting, σ are symmetric functions of the family of shape operators u (Ae1,...,Aeq) corresponding to the choice of the orthonormal basis (e ) in the α normal bundle to L ⊂ M. Thus σ are functions on the principal bundle P of all u orthonormal frames over the submanifold. The integrals of σ over P with respect u to normalized measure are called total extrinsic curvatures and denoted by σˆ . u Notice, that in the case of distributions and foliations diﬀerent types of cur- vatures has been recently considered [21, 20], mainly, in order obtain integral formulas. The ones used in this context have been introduced by K. Andrzejew- ski, W. Koz lowski and the author [3] for distributions, not to study generalized minimality, but to obtain diﬀerent types of integral formulas related to extrinsic geometry. In this article, we derive the formula for the variation dσˆ . We show that at dt u t = 0 d σˆ = g(R −S ,V)dvol , u u u L dt ZL where V is the variation ﬁeld and R ,S are integrals over the ﬁbers of the u u following sections of the normal bundle to L over P, R = tr((R(e ,·)e ) T )e , S = tr(A T )e , u α β ⊤ α♭(u) β u α u α α,β α X X respectively. Thus the condition of u–minimality equals R = S (Theorem 5.6). u u In the case of M being a space form with sectional curvature c, R reduces to u c(n−|u|+1)H , where u Hu = σα♭(u)eα. α X These results generalize the codimension one results by Reilly [17] and, in a sense, the results of Cao and Li [10]. This is due to the fact that T in codimension one u reduces to the classical Newton transformation T , whereas transformations T in r r arbitrarycodimensionusedin[10]areobtainedfromT formulti–indices oflength u |u| = r (see [3]). In the end, we give some examples of u–minimal submanifolds. 4 K.NIEDZIAL OMSKI Among others, we show that 0–minimality is equivalent to classical notion of minimality. Throughout the paper, we use the following index convention: i,j,k = 1,2,...,dimL; α,β,γ = q +1,q +2,...,dimM. 2. Generalized Newton transformation and its basic properties In [3] the authors introduced the notion of the generalized Newton transfor- mation in order to study geometry of foliations of codimension higher than one. This transformation generalizes the classical Newton transformation to the case of ﬁnite family of operators. Let us recall this notion and state its properties. Fix a positive integer q and let u ∈ Nq (N denotes here the set of all non– negative integers). We deﬁne the length of u by |u| = u + u + ...,+u . Let 1 2 q A = (A ,A ,...,A ) be a family of square n by n operators (matrices) on some 1 2 q vector space V. Then we can consider the characteristic polynomial χA of the form χA(t) = det(I +tA), where t = (t ,...,t ) and tA = t A +t A +...+t A . Clearly, 1 q 1 1 2 2 q q χA(t) = σu(A)tu, u n |X|≤ for some constants σ = σ (A), where tu = tu1tu2...tuq for u = (u ,...,u ). We u u 1 2 q 1 q call σ the symmetric functions associated with the system A. u Deﬁnition 1. A system T = T (A), |u| ≤ n, of endomorphisms is called the u u generalized Newton transformation if the following condition holds: Let A(t) be a one parameter family of operators such that A(0) = A and let σ (t) be the u corresponding symmetric functions. Then d d (2.1) σ (t) = tr A (t) ·T , dt u t=0 dt α t=0 α♭(u) α (cid:18) (cid:19) X where α (u) denotes the multi–index obtained from u by subtracting 1 in the ♭ α–th coordinate. One can show that (T ) exists and is unique [3]. Moreover, it can be u u n | |≤ characterized by the following recursive properties T = I, 0 = (0,0,...,0), 0 (2.2) T = σ I − A T u u α α♭(u) α X = σ I − T A , |u| ≥ 1. u α♭(u) α α X The generalized Newton transformation has the following useful properties [3]. Proposition 2.1. Let A = (A ,...,A ) be a family of operators on a vector 1 q space V. Let (T ) be the generalized Newton transformation associated with A u FRAME BUNDLE APPROACH TO GENERALIZED MINIMAL SUBMANIFOLDS 5 and let σ be the corresponding symmetric functions. Then the following relations u hold |u|σ = tr(A T ), u α α♭(u) α X trT = (n−|u|)σ . u u One can obtain the direct formula for T and for σ (see [3], eq. (3) and (10)). u u In the Section 6 devoted to examples we will need the following algebraic relation. Let us ﬁrst recall necessary notation. For a multi–index u ∈ Nq and p = |u| let p p! = . u u !...u ! (cid:18) (cid:19) 1 q Moreover, we write v ≤ u for two mulit–indices u,v,∈ Nq if u−v ∈ Nq. Proposition 2.2. Let A be an endomorphisms of n–dimensional vector space V. Let A = (ρ A + µ I,...,ρ A + µ I) be a system of endomorphisms, where 1 1 q q ρ = (ρ ,...,ρ ) and µ = (µ ,...,µ ) are ﬁxed elements of Rq. Then, for any 1 q 1 q multi–index u ∈ Nq, 1 n−|u|+|v| |u|−|v| (2.3) σ (A) = ρu vµvσ (A). u − u v (n−|u|)! v u−v | |−| | v u(cid:18) (cid:19)(cid:18) (cid:19) X≤ Proof. Assume ﬁrst that A = (A +µ I,A ,...,A ) and A = (A ,A ,...,A ) 1 1 2 q 0 1 2 q for some endomorphisms A1,A2,...,Aq. Then the characteristic polynomial χA equals χA(t) = det(I +t1(A1 +µI)+t2A2 +...+tqAq) = det((1+t µ)I +tA ) 1 0 t = (1+t µ)ndet I + A 1 1+t µ 0 (cid:18) 1 (cid:19) = (1+t µ)n σ (A )(1+t µ) utu 1 u 0 1 −| | u X = σ (A )(1+t µ)n utu u 0 1 −| | u X n u −| | n−|u| = σ (A ) µjtjtu. u 0 j 1 u j=0 (cid:18) (cid:19) X X Thus u1 n−|u|+j σ (A) = µjσ (A ). u j 1j(u) 0 ♭ j=0 (cid:18) (cid:19) X Applying above formula consecutively to each ’coordinate’ we get 1 n−|u|+|v| (2.4) σ (A +µ I,...,A +µ I) = µvσ (A ). u 1 1 q q (n−|u|)! v u−v 0 v u(cid:18) (cid:19) X≤ 6 K.NIEDZIAL OMSKI By the formula σ (A,A,A ,...,A ) = u1+u2 σ (A,A ,...,A ) (see u 3 q u1 (u1+u2,u3,...,uq) 3 q [21]) we have (cid:0) (cid:1) |u| (2.5) σ (A,A,...,A) = σ (A). u u u | | (cid:18) (cid:19) Finally,(2.4),(2.5)andtheformulaσ (aA ,...,A ) = au1σ (A )imply(2.3). (cid:3) u 1 q u 0 3. Frame bundle approach to submanifold geometry Let (M,g) be a Riemannian manifold, L a codimension q immersed subman- ifold of M, i.e., we have a surjective immersion ϕ : L → M. The Riemannian metric on L is such that ϕ becomes an isometry and, therefore, we often identify L with its image ϕ(L) ⊂ M. Consider the pull–back bundle ϕ 1TM over L with − the ﬁber (ϕ 1TM) = T M, p ∈ L. − p ϕ(p) In other words, ϕ 1TM is a tangent bundle of M restricted to L. There is the − unique connection ∇ϕ in this bundle satisfying the following condition (see, for example, [6]) ∇ϕϕ Y = ∇ Y, X,Y ∈ Γ(TL), X X ∗ where ∇ is the Levi–Civita connection on L. The bundle ϕ 1TM splits into two bundles ϕ 1T L and ϕ 1T Lwith respect − − ⊤ − ⊥ to the decomposition T M = T ϕ(L)⊕(T ϕ(L)) , p ∈ L. ϕ(p) ϕ(p) ϕ(p) ⊥ Denote by ∇ϕ, and ∇ϕ, the induced connections in these bundles, respectively. ⊤ ⊥ Notice that ϕ 1T L is isomorphic to TL via ϕ . − ⊤ ∗ ϕ 1T L ⊕ ϕ 1T L ϕ 1TM (cid:31)(cid:127) // TM − ⊤ − ⊥ − (cid:15) (cid:15) (cid:15) (cid:15) / ϕ / TL / L / M Fix a local section N ∈ Γ(ϕ 1T L). Then N induces a shape operator AN : − ⊥ TL → TL by the formula ϕ AN(X) = (∇ϕN)⊤, X ∈ TL. X ∗ Thus A ∈ Γ(End(TL)). Let now P = O(ϕ 1T L) (or P = SO(ϕ 1T L) assuming L is transver- − ⊥ − ⊥ sally orientable) be a bundle of (oriented) orthonormal frames of ϕ 1T L. Al- − ⊥ ternatively, Γ(P) consists of all sections, which assign an orthonormal basis (p,e) = (e ) , where e ∈ T L, to any point p ∈ L. For a ﬁxed element α α=1,...,q α ϕ⊥(p) (p,e) of P we may consider a family of shape operators A(p,e) = (Ae1,Ae2,...,Aeq). Toeachsuchfamilywecanassociatethesymmetricfunctionsσ (p,e) = σ (A(p,e)), u u where u = (u ,u ,...,u ) ∈ Nq is a ﬁxed multi–index. Thus σ ∈ C (P). 1 2 q u ∞ FRAME BUNDLE APPROACH TO GENERALIZED MINIMAL SUBMANIFOLDS 7 Each ﬁber P , p ∈ L, can be identiﬁed with the structure group G = O(q) p (or G = SO(q), respectively). Considering a (normalized) Haar measure on G we deﬁne an integral of any smooth function f : P → R by f dvol = f(e g)dg, Px 0 ZPp ZG where e is a ﬁxed element of P . One can show that this deﬁnition is independent 0 p of the choice of e . Therefore σ induces a smooth function σˆ on L, 0 u u σˆ (p) = σ(p,e)dvol (p,e), p ∈ L. u Pp ZPp We call σˆ the generalized extrinsic curvature of L or u–extrinsic curvature. For u some choices of u the generalized extrinsic curvatures vanish, but there are many multi–indices uforwhichσˆ isnon–zeroandgives informationonthesubmanifold u (see the Section 7 – Final Remarks). Moreover, let π 1TM be a vector bundle over P with the ﬁber P− (π 1TM) = T M. P− (p,e) ϕ(p) This bundle splits into direct (orthogonal) sum of bundles E and E with respect ′ to the decomposition T M = T L⊕(T L) . In other words, ϕ(p) ϕ(p) ϕ(p) ⊥ E = (ϕ 1T L) , E = (ϕ 1T L) . (p,e) − ⊤ p (′p,e) − ⊥ p The connection ∇ϕ induces, with respect to the above decomposition, two con- nections ∇E and ∇E′ in bundles E and E , respectively. In particular, we have ′ ′ ∇EWA αY = ∇EWAα Y −A∇EWeαY ′ (cid:0) (cid:1) = (cid:0)∇EW(AαY(cid:1))−Aα(∇EWY)−A∇EWeαY, where we treat e as a section of E . Moreover, recall the Codazzi formula α ′ (3.1) (Rϕ(X,Y)N) = (∇EA)NX −(∇EA)NY, X,Y ∈ TP, N ∈ E . ⊤ Y X ′ The key fact about diﬀerentiation of symmetric functions σ in the horizontal u direction follows by the deﬁnition (2.1) of generalized Newton transformation and is given below. Denote by H the horizontal distribution in P induced by the connection ∇ϕ, . ⊥ Proposition 3.1. The following formula holds (3.2) Wσ = tr ∇E A ·T , W ∈ H. u W α α♭(u) α X (cid:0)(cid:0) (cid:1) (cid:1) Proof. Fix a basis (p,e) and let W ∈ H . Then W = Xh for some X ∈ T L. (p,e) (p,e) p Let γ be a curve on L such that γ(0) = p and γ˙(0) = X. Choose a local section s of P parallel at (p,e). Then (see the Appendix) at (p,e) we may write (∇EWA)αY = ∇ϕX,⊤(AeαY ◦s)−Aeα(∇ϕX,⊤(Y ◦s))−A∇ϕX,⊥eα(Y ◦s). Consider a family B(t) = (B (t),...,B (t)) of endomorphisms of T L given by 1 q p the formula Bα(t) = (τt⊤)−1 ◦Aτt⊥(eα) ◦τt⊤, 8 K.NIEDZIAL OMSKI where τ : T L → T L and τ : T L → T L denote the parallel transports t⊤ p γ(t) t⊥ p⊥ γ⊥(t) along γ in TL and T L, respectively. Since we compute dσ (γh(t)) , then ⊥ dt u t=0 we consider bases (e ) along γh(t). Therefore (e ) at γh(t) is a parallel trans- α α port of (e (p)). Hence τ (e (p)) = e (γh(t)) and B (t) = A (γh(t)). Moreover, α t⊥ α α α α dB (t) = (∇E A) , thus by (2.1) we obtain (3.2). (cid:3) dt α t=0 W α Let us deﬁne for a ﬁxed multi–index u ∈ Nq three sections R ,H ,S ∈ Γ(E ). u u u ′ Namely, we put R (p,e) = − tr(R T )e , u αβ α♭(u) β α,β X (3.3) H (p,e) = σ e , u α♭(u) α α X S (p,e) = tr(A T )e , u α u α α X where R : P → End(ϕ 1T L) is given by αβ − ⊤ R (X) = (R(e ,X)e ) . α,β α β ⊤ These sections induce sections R , H and S of ϕ 1T L, respectively, by u u u − ⊥ R (p) = R (p) = R (p,e)dvol (p,e), u u u Pp ZPp c (3.4) H (p) = H (p) = H (p,e)dvol (p,e), u u u Pp ZPp c Su(p) = Su(p) = Su(p,e)dvolPp(p,e). ZPp The above integrals make secnse and are well deﬁned (see the Appendix). Remark 3.2. Notice that S is deﬁned in the same way as S by Cao and Li u r+1 [10], whereas we are not aware of the existence of the analogue of H (or H ) u u in the literature, except for [3]. In [3] the authors computed divergence of the section H +W , where u u Wu(p,e) = Tα♭β♭(u)(p,e)(∇eαeβ)⊤p . α,β X and obtained the family of integral formulas, which generalize many formulas relating intrinsic and extrinsic geometry of foliations known in the literature. In the codimension one case, (3.5) S = (u+1)σ N, and H = σ N, u u u u 1 − where N is a unit normal vector to L. 4. Properties of some differential operators Consider the notation from the previous section. In this section we deﬁne two diﬀerential operators, which will appear in the main considerations. These results are an adaptation of Rosenberg’s results [19] concerning the classical Newton transformation to the case of the generalized Newton transformation. FRAME BUNDLE APPROACH TO GENERALIZED MINIMAL SUBMANIFOLDS 9 Firstly, for any section W ∈ Γ(E ) denote by W a function on P of the form ′ α W (p,e) = g(W,e ) , (p,e) ∈ P. α α p Fix a muliti–index u ∈ Nq and consider a diﬀerential operator L : Γ(E ) → u ′ C (P) ∞ L (W) = tr(((∇E)2W) ·T ), W ∈ Γ(E ). u α α♭(u) ′ α X In other words, L (W) = g((∇E)2 W),e )(T ) , u ei,ej α α♭(u) ji α,i,j X where (e ) is an orthonormal basis on L and B denotes the coeﬃcients of the i ij operator B : TL → TL with respect to this basis. Let us derive the formula for the value of L on sections of the form V ◦ π , where V ∈ Γ(ϕ 1T L). Notice u P − ⊤ that the deﬁnition of L is tensorial with respect to e . Thus, for a ﬁxed point p, u α extending (e ) locally to a parallel basis at p, we may write L as α u L (V ◦π ) = g(∇E∇E (V ◦π ),e ) u P p ehi (Tα♭(u)ei)h P α α,i X = g(∇ϕei,⊥∇ϕTα,⊥♭(u)eiV,eα) α,i X = div(g(∇ϕTα,⊥♭(u)eiV,eα)ei), α,i X where we also assume that (ei) is parallel at p. The quantity g(∇ϕTα,⊥♭(u)eiV,eα)ei does not depend on the choice of orthonormal basis (e ) and, therefore, is well i deﬁned section of E. In particular, it follows that (compare the Proposition 8.2 and the formula (8.2) in the Appendix) (4.1) L (V ◦π )dvol = 0. u P P ZP Since, for any function f ∈ C (L) ∞ g(∇ϕTα,⊥♭(u)eiV,eα)(eif) = g(∇ϕTα,⊥♭(u)(∇f)V,eα) = Tα♭(u)(∇f)g(V,eα), α,i α α X X X we get (4.2) fL (V ◦π )dvol = − (T (∇f))hV dvol . u P P α♭(u) α P ZP α ZP X We will compute the integral of the right hand side of (4.2). Let us begin with preliminary results. These results are generalizations of the ones obtained by Rosenberg [19]. Proposition 4.1. Let Y ∈ Γ(ϕ 1T L). The following relation holds − ⊤ (4.3) g(∇E T (Y),e ◦π ) = g(∇E (Y ◦π ),e ◦π )+K (Y), ehi u i P Tu(ei)h P i P u i i X X 10 K.NIEDZIAL OMSKI where K (Y) is given by the following recursive relation u Ku(Y) = − Kα♭(u)(AαY)− g(Rϕ(Y,Tα♭(u)ei)eα,ei)− g(A∇ϕTα,⊥♭(u)eαY,ei) α i,α i,α X X X and K (Y) = 0. 0 Proof. First, notice that the above relation is tensorial with respect to Y. Fix a basis (p,e) ∈ P and consider a local section σ of P parallel at (p,e). Then the formula (4.3) can be written at a point (p,e) as g(∇ϕei,⊤(Tu(Y)◦σ),ei) = g(∇ϕTu,⊤(ei) σY,ei)+Ku(Y). ◦ i i X X We will use induction. Assume that the above formula holds for all multi–indices v of length |v| < |u|. We may assume Y is parallel at p with respect to ∇ϕ, . ⊤ Thus, it suﬃces to show that (4.4) g(∇ϕ, T (Y),e ) = K (Y). ei⊤ u i u i X By the recursive deﬁnition of T , Codazzi equation (3.1), formula (3.2) and in- u ductive assumption, we have g(∇ϕ, T (Y),e ) = Y(σ ◦π )− g(∇ϕ, T A Y,e ) ei⊤ u i u P ei⊤ α♭(u) α i i i,α X X = Y(σu ◦πP)− g((∇ϕTα,⊤♭(u)eiAα)Y,ei)− Kα♭(u)(AαY) i,α α X X = Yhσ − g((∇ A) T e ,e ) u Y α α♭(u) i i i,α X − g(Rϕ(Y,T e )e ,e ) α♭(u) i α i i,α X − g(A∇Tϕα,⊥♭(u)eieαY,ei)− Kα♭(u)(AαY) i,α α X X = K (Y), u (cid:3) which proves (4.4). In the Proposition below div denotes the divergence of sections of the bundle E E and Hess (ϕ) is a Hessian of ϕ ∈ C (P) in the horizontal direction (see the ∞ H Appendix for more details). Proposition 4.2. The following relation holds div (T (∇f)) = tr(Hess (f ◦π )·T )+K (∇f), f ∈ C (L), E u P u u ∞ H for any multi–index u ∈ N(q).

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.