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EXTERIOR DIFFERENTIAL FORMS ON RIEMANNIAN SYMMETRIC SPACES
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EXTERIOR DIFFERENTIAL FORMS ON RIEMANNIAN SYMMETRIC SPACES
Alexandrova Irina 1
Stepanov Sergey 2
Tsyganok Irina 3
Authors:
1.  Finance University under the Government of Russian Federation

Russian Federation
2.  Finance University under the Government of Russian Federation

Russian Federation
3.  Finance University under the Government of Russian Federation

Russian Federation
Type:
Article
DOI:
https://doi.org/10.21603/2500-1418-2017-2-2-49-53
Pages:
from 49 to 53
Status:
Published
Received:
30.12.2017
Accepted:
30.12.2017
Published:
30.12.2017
Subject area:
GRNTI 29.01 Общие вопросы физики
GRNTI 27.01 Общие вопросы математики
GRNTI 31.01 Общие вопросы химии
GRNTI 34.01 Общие вопросы биологии
Language:
English
Keywords:
Riemannian symmetric space, conformal Killing L2-form, harmonic form
Abstract and keywords
Abstract (English):
In the present paper we give a rough classification of exterior differential forms on a Riemannian manifold. We define conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms due to this classification and consider these forms on a Riemannian globally symmetric space and, in particular, on a rank-one Riemannian symmetric space. We prove vanishing theorems for conformal Killing L 2-forms on a Riemannian globally symmetric space of noncompact type. Namely, we prove that every closed or co-closed conformal Killing L 2-form is a parallel form on an arbitrary such manifold. If the volume of it is infinite, then every closed or co-closed conformal Killing L 2-form is identically zero. In addition, we prove vanishing theorems for harmonic forms on some Riemannian globally symmetric spaces of compact type. Namely, we prove that all harmonic one-formsvanish everywhere and every harmonic r -form  r  2 is parallel on an arbitrary such manifold. Our proofs are based on the Bochnertechnique and its generalized version that are most elegant and important analytical methods in differential geometry “in the large”.

Keywords:
Riemannian symmetric space, conformal Killing L2-form, harmonic form
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INTRODUCTION In the present paper we consider conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms which are defined on Riemannian differential r-forms and taking their values in the space of homogeneous tensor fields on M, g. Bourguignon has proved the existence of three basis natural operators of this space, but only the following globally symmetric spaces. In particular, we prove two D1 and D2 of them were recognized. The first vanishing theorems for conformal Killing, closed conformal Killing and coclosed conformal Killing operator D1 is the exterior differential operator L2-forms on Riemannian globally symmetric spaces of d : Cr M  Cr1M and the second operator noncompact type. In addition, we prove vanishing D2 is the exterior co-differential operator theorems for harmonic forms on some Riemannian globally symmetric spaces of compact type. Our proofs are based on the Bochner technique and its generalized d  : C r M  C r 1M . About the third basis natural operator D3 , it was said version that are most elegant and important analytical methods in differential geometry “in the large”. The results of the present paper were announced at the International Conference "Differential Geometry" organized by the Banach Center from June 18 to June 24, 2017 at Będlewo (Poland) and at the International Conference "Modern Geometry and its Applications" dedicated to the 225th anniversary of the birth of N.I. Lobachevsky and organized by the Kazan Federal that except for case r = 1, this operator does not have any simple geometric interpretation. Next, for the case r = 1, it was explained that the kernel of this operator consists of infinitesimal conformal transformations on M , g . In connection with this, we have received a specification of the Bourguignon proposition and proved (see [4]) that the basis of natural differential operators consists of three operators of following forms: University from November 27 to December 3, 2017 at Kazan (Russian Federation). D  d ; 1 1 r1 D  g  d ; 1  2 nr1 PRELIMINARIES D3    1 r1 d  1 nr1 g  d  More then thirty years ago Bourguignon has where   investigated (see [1]) the space of natural (with respect g  d  r X 0 , X 1 ,..., X r   to isometric diffeomorphisms) differential operators of  1a gX , X d  X ,..., X , X ,..., X  order one determined on vector bundle r M of exterior    a  2 0 a  1 a 1 a 1 r Please cite this article in press as: Alexandrova I.A., Stepanov S.E., and Tsyganok I.I. Exterior differential forms on riemannian symmetric spaces. Science Evolution, 2017, vol. 2, no. 2, pp. 49-53. DOI: 10.21603/2500-1418-2017-2-2-49-53. Copyright © 2017, Alexandrova et al. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix, transform, and build upon the material for any purpose, even commercially, provided the original work is properly cited and states its license. This article is published with open access at http:// science-evolution.ru/. 49 for an arbitrary exterior differential r-form  and any Hr(M, ℝ)  Dr(M, ℝ)  Fr(M, ℝ). On a closed and vector fields X1, X 2 ,..., Xr on M. oriented Riemannian manifold M , g   the condition The kernel of D1 consists of closed exterior   ker d  ker d is equivalent to the following differential r-forms, the kernel of D2 consists of cocondition   ker  (see, for example, [37 pp. 202]). closed exterior differential r-forms and kernel of D3 consists of conformal Killing r-forms or, in other words, conformal Killing-Yano tensors of order r that constitute three vector spaces Dr(M, ℝ), Fr(M, ℝ) and Tr(M, ℝ) respectively. These vector spaces is subspaces of the vector space of exterior differential r-forms on M , g  which we denote by r(M, ℝ). Remark. The concept of conformal Killing tensors Remark. Harmonic forms are a classical object of investigation of differential geometry for the last seventy years, beginning with the well-known scientific works of Hodge (see, for example, [24, pp. 107-113] and [25]). It is well known by Hodge that the vector space Hr(M, ℝ) of harmonic r-forms on a compact n-dimensional (1  r  n - 1) Riemannian manifold M,g  is finitedimensional. Moreover, the dimension of Hr(M, ℝ) was introduced by S. Tachibana about forty years ago equals to the Betti number br M  of M,g such that (see [5]). He was the first who has generalized some br M   b n r M  (see [2, pp. 202-208; 385-391]). results of a conformal Killing vector field (or, in other words, an infinitesimal conformal transformation) to a skew symmetric covariant tensor of order 2 named him the conformal Killing tensor. Kashiwada has generalized this concept to conformal Killing forms of order In conclusion, we denote by Cr(M, ℝ) the vector space of parallel or covariantly constant r-forms on M with respect to  , i.e. Cr(M, ℝ)  Tr(M, ℝ)  Dr(M, ℝ)  Fr(M, ℝ). r  2 (see [18]). The theory of conformal Killing forms Riemannian symmetric spaces are also a classical is contained in the monographs [9] and [15]. In addition, there are many various applications of these tensors in theoretical physics (see, for example, [3]; [4]; [9]; [10]; [20]; [21]; [22]). In particular, we have proved that the vector space Tr(M, ℝ) on a compact n-dimensional object of investigation of differential geometry as differential forms. Cartan obtained the basic theory of symmetric spaces between 1914 and 1927. Riemannian symmetric spaces have been studied by many others authors. In particular, beginning in the 1950s, Harish- (1  r  n - 1) Riemannian manifold M,g  is finiter Chandra, Helgason, and others developed harmonic analysis and representation theory on these spaces and dimensional. In addition, the number t r M   T (M, ℝ) their Lie groups of isometries (see, for example, is a conformal invariant M of M,g tr M   tn r M . such that [2, pp. 235-264]; [13]; [17]; [26]; [38]; [39, pp. 222-292]). We recall here that the Riemannian symmetric space The condition   ker D3  ker D2 characterizes an is a finite dimensional Riemannian manifold M,g , such r-form  as co-closed conformal Killing form. Therefore, that for every its point x there is an involutive geodesic the space of all co-closed conformal Killing r-forms we can define as Kr(M, ℝ)  Tr(M, ℝ)  Fr(M, ℝ). A co- closed conformal Killing form is also called as the symmetric sx . M,g sx , such that x is an isolated fixed point of is said to be Riemannian locally symmetric if Killing-Yano tensor (see, for example, [9, pp. 426-427]; [10; pp. 559-564]). its geodesic symmetries are in fact isometric. This is equivalent to the vanishing of the covariant derivative of In turn, the condition   ker D1  ker D2 the curvature tensor R of M,g (see [39, p. 244]). characterizes the form  as a closed conformal Killing form or closed conformal Killing-Yano tensor (see, for example, [9, pp. 416]; [23]). Sometimes, closed conformal Killing forms are also called planar forms (see [16]). Therefore, the space of all closed conformal Killing r-forms we can define as Pr(M, ℝ)  Tr(M, ℝ)  Dr(M, ℝ). Remark. The concept of Killing tensors was introduced by K. Yano about fifteen years ago (see [19]). He was the first who has generalized some results of a Killing vector field (or, in other words, an infinitesimal isometric transformation). In turn, planar forms generalized the concept of concircular vector fields. In addition, we have proved that the vector spaces Kr(M, ℝ) and Pr(M, ℝ) on a compact n-dimensional (1  r  n - 1) Riemannian manifold M,g  are finite- A Riemannian locally symmetric space is said to be a Riemannian globally symmetric space if, in addition, its geodesic symmetries are defined on all M,g . A Riemannian globally symmetric space is complete (see [39, p. 244]). In addition, a complete and simply connected Riemannian locally symmetric space is a Riemannian globally symmetric space (see [39, p. 244]). Riemannian globally symmetric spaces can be classified by classifying their isometry groups. The classification distinguishes three basic types of Riemannian globally symmetric spaces: spaces of so- called compact type, spaces of so-called non-compact type and spaces of Euclidean type (see, for example, [26; p. 207-208]; [39, p. 252]). An addition, if M,g is a Riemannian globally symmetric spaces of compact dimensional. Moreover, the numbers kr M   Kr(M, type then M,g  is a compact Riemannian manifold with ℝ) and r pr M   P (M, ℝ) are conformal invariant of non-negative sectional curvature and positive-definite M,g  such that kr M   pn r M  (see [15] and [16]). Ricci tensor (see [39, p. 256]). The condition   ker D1  ker D2 characterizes the Let M,g  be a Riemannian locally symmetric space, form  as a harmonic form (see [19]; [24, pp. 107-113]). then  Ric  0 for the Ricci tensor Ric of M,g . If, in Hence, the space of all harmonic r-forms we can define as addition, M,g  is irreducible, then it is Einstein (see [2, p. 254]). If in this case, the Einstein constant is Volg M  of M,g is infinite then we obtain a positive, then it follows from Myer’s theorem (see 2 r [2, p. 171]) that the space M,g is compact. Then one can contradiction with our condition that ω  L Λ M . show that the curvature operator is nonnegative (see also It remains to recall that a Riemannian globally [2, p. 254]). Therefore, M,g  is a Riemannian globally symmetric spaces of non-compact type M,g  is a symmetric space of compact type. If, in addition, M,g  is simply-connected and its curvature operator is positive- definite, then it is a Euclidian sphere (see also [2, p. 228]). Therefore, we can conclude that a simply-connected and irreducible Riemannian locally symmetric space with positive-definite curvature operator is a Euclidian sphere. complete non-compact Riemannian manifold with nonpositive sectional curvature. If a Riemannian symmetric space of non-compact type has an even dimension then the following theorem on conformal Killing L2 -forms is true. Theorem 2. Let M,g  be a 2r- dimensional m  2 On the other hand, if M,g  is a Riemannian simply connected Riemannian symmetric space of non- 2 globally symmetric spaces of non-compact type then compact type. Then every conformal Killing L -form of M,g  is a complete non-compact Riemannian degree r is parallel form on M,g. If the volume of manifold with non-positive sectional curvature and negative-definite Ricci tensor, and diffeomorphic to a M,g is infinite, then every conformal Killing of degree r is identically zero on M, g  . L2 -form Euclidean space (see [39, p. 256]). Proof. In our paper [31] we have proved that an We know that an irreducible Riemannian locally arbitrary conformal Killing L2 -form of degree r is symmetric space M,g  is Einstein (see [2, p. 254]). parallel form on 2r- dimensional complete non-compact If the Einstein constant is negative, then it follows from Riemannian manifold M,g with negative semi-define Bochner’s theorem on Killing fields that the space is noncompact (see [2, p. 254]). In this case, one can curvature operator. It means that Cr(M, ℝ)  Tr(M, ℝ). If show that the curvature operator is nonpositive (see the volume of M,g is infinite, then every conformal also [2, p. 254]). Therefore, M,g is a Riemannian Killing L2 -form of degree r is identically zero on M,g . globally symmetric space of noncompact type. CONFORMAL KILLING FORMS ON RIEMANNIAN GLOBALLY SYMMETRIC SPACES In this section we consider the natural Hilbert space In this case, Theorem 2 is a corollary of this proposition. An important invariant of a symmetric space is its rank which is the maximal dimension of a totally geodesic flat subspace. In particular, rank-one symmetric spaces are an important class among symmetric spaces (see [16] and [17]). In his book [17], Chavel gave a L2 Λr M  C Λr M  of L2 -forms on a complete beautiful account of the rank-one symmetric spaces from noncompact Riemannian manifold M,g  determined by the condition 2 which is a geometric point of view up to the classification of them, which he left for the reader to pursue as a matter in Lie group theory. We prove the following theorem for conformal Killing forms on real rank-one symmetric  ω d Volg   M spaces using his results. Theorem 3. Let M,g  be a simply connected for an arbitrary form ω  C  Λr M . First, we prove the following theorem for closed and symmetric space of rank one type. If, in addition, M,g  is a space of compact type, and of odd dimension coclosed conformal Killing L2 -forms on a Riemannian n  2k  1 for k  1 , symmetric space of noncompact type that is a complete and simply connected Riemannian manifold of non- positive sectional curvature. Side by side, its curvature then tr M   n  2! r  1!n  r  1! , kr M   n  1!r  1!n  r! operator R is nonpositive (see [39]). and pr M   n  1! r!n  r  1! Theorem 1. Let M,g  be an n-dimensional (n  3) simply connected Riemannian symmetric space of non- compact type. Then every closed or co-closed conformal for an arbitrary 1  r  2k . Proof. We consider a rank-one Riemannian symmetric space M,g  of compact type. If, in addition, Killing L2 -form on M,g  is a parallel form. If the M,g  is a simply connected manifold of odd dimension volume of M,g  is infinite, then every closed or con  2k  1 then M,g  is a sphere of constant sectional closed conformal Killing identically zero. L2 -form on M,g  is curvature (see [13]). The vector spaces over ℝ of conformal Killing, coclosed and closed conformal Proof. In our paper [31] we have proved that an arbitrary closed (resp. coclosed) conformal Killing L2 -form  is parallel on a complete noncompact Riemannian manifold M,g  with nonpositive curvature Killing p-forms on an Euclidian n-sphere have finite dimensions which are equal to tr M   n  2! r  1!n  r  1!, kr M   n  1!r  1!n  r! operator. It means that Cr(M, ℝ)  Pr(M, ℝ)  Kr(M, ℝ). and pr M   n  1!r!n  r  1! , respectively (see [15] In this case,  2  const . If we suppose that the volume and [16]). This proves our Theorem 3. HARMONIC FORMS ON A RIEMANNIAN M,g is a manifolds of compact type, and of odd GLOBALLY SYMMETRIC SPACES dimension, then its Betti numbers b M   b M   1, In this section we consider harmonic forms on a b M   ...  b M   0 0 n M,g Riemannian globally symmetric space of compact type. 1 n1 . On the other hand, if is In the simply connected case, “compact type” is equivalent to the compactness condition that was a simply connected symmetric space of rank one, and of non-compact type, then M,g carries no harmonic considered in Section 4 in [41]. First we prove the following theorem for harmonic L2-forms except when is infinite dimensional. r  n 2 in which case Hr(M, ℝ) forms on a Riemannian symmetric space of compact type that is a compact Riemannian manifold with non-negative sectional curvature and positive-definite Ricci tensor. Side Proof. Chavel has proved in [17] that if M,g  is a rank-one Riemannian symmetric space of compact type n n n n by side, its curvature operator R is nonnegative then it is one of the four spaces: S , ℝP , CP 2 , HP and (see [35]). In this case, the Bochner technique tells us that all harmonic r-forms (2  r  n - 1) are parallel and OP . The final is the 16-dimensional Cayley plane. Accordingly, compact-type symmetric spaces of rank- one have strictly positive sectional curvature. In the case 1-forms are identically zero (see [2, pp. 208; 212; 221]). Now, a parallel form is necessarily invariant under the of odd dimension M,g is a Euclidian sphere holonomy. Thus, we are left with a classical invariance (see [16]). On the other hand, it is well known that a problem (see [37, pp. 306-307]). In this case, the Betti 1 compact Riemannian manifold M,g  with positive numbers b1 M   bn1 M   dim H (M, ℝ)  0 and constant sectional curvature admits no nonzero harmonic r  n  forms (see [2, p. 212]). Therefore, its Betti numbers br M   dim H (M, ℝ)    for any r  2,..., n  2 b1 M   ...  bn1 M   0 and b0 M   bn M   1.  r  On the other hand, if M,g  is a real Riemannian (see [2, p. 212]). Since there is exactly one harmonic 0-form (a constant) on compact manifolds and the dual symmetric space of non-compact type, and of rank one then it is one of the four spaces: Hn, CHn , HHn , OH2 , n-form then the Betti numbers proved the following theorem. b0 M   bn M   1. We i.e., real hyperbolic space, complex hyperbolic space, quaternionic hyperbolic space and the octonionic Theorem 4. Let (M, g) be an n-dimensional (n  2) non-compact simply connected Riemannian globally symmetric space of compact type. Then all harmonic hyperbolic plane (see [42]; [43]). We can normalize their Riemannian metrics so that the maximum of the sectional curvature is -1 (see also [42]; [43]). In this one-forms vanish everywhere and every harmonic r-form case, Dodziuk has proved that M,g  carries no (r  2) is parallel. In this case, the Betti numbers  n  harmonic L2-forms except when r r  n 2 in which case b1 M   bn1 M   0 r  2,..., n  2 . and r br M       for an arbitrary H (M, ℝ) is infinite dimensional (see [32]). This completes the proof of Theorem 5. In conclusion, we can formulate an obvious statement. Theorem 5. Let M,g be an n-dimensional simply connected symmetric space of rank one. If, in addition,
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