EXTERIOR DIFFERENTIAL FORMS ON RIEMANNIAN SYMMETRIC SPACES

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-forms on a Riemannian globally symmetric space of noncompact type. Namely, we prove that every closed or co-closed conformal Killing L-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-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-forms vanish everywhere and every harmonic r-form   2  r is parallel on an arbitrary such manifold. Our proofs are based on the Bochner technique and its generalized version that are most elegant and important analytical methods in differential geometry “in the large”.


INTRODUCTION
In the present paper we consider conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms which are defined on Riemannian globally symmetric spaces.In particular, we prove vanishing theorems for conformal Killing, closed conformal Killing and coclosed conformal Killing L 2 -forms on Riemannian globally symmetric spaces of noncompact type.In addition, we prove vanishing theorems for harmonic forms on some Riemannian globally symmetric spaces of compact type.Our proofs are based on the Bochner technique and its generalized 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 University from November 27 to December 3, 2017 at Kazan (Russian Federation).

PRELIMINARIES
More then thirty years ago Bourguignon has investigated (see [1]) the space of natural (with respect to isometric diffeomorphisms) differential operators of order one determined on vector bundle : . About the third basis natural operator 3 D , it was said 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 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: where for an arbitrary exterior differential r-form  and any vector fields r X X X ..., , , 2 1 on M. The kernel of 1 D consists of closed exterior differential r-forms, the kernel of 2 D consists of co- closed exterior differential r-forms and kernel of 3 D consists of conformal Killing r-forms or, in other words, conformal Killing-Yano tensors of order r that constitute three vector spaces D r (M, ℝ), F r (M, ℝ) and T r (M, ℝ) respectively.These vector spaces is subspaces of the vector space of exterior differential r-forms on   g M , which we denote by  r (M, ℝ).
Remark.The concept of conformal Killing tensors was introduced by S. Tachibana about forty years ago (see [5]).He was the first who has generalized some 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 2  r (see [18]).The theory of conformal Killing forms 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 T r (M, ℝ) on a compact n-dimensional  [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 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 K r (M, ℝ) and P r (M, ℝ) on a compact n-dimensional (see [15] and [16]).The condition 2 1 ker ker D D    characterizes the form  as a harmonic form (see [19]; [24, pp. 107-113]).
Hence, the space of all harmonic r-forms we can define as 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 H r (M, ℝ) of harmonic r-forms on a compact n-dimensional is finitedimensional.Moreover, the dimension of H r (M, ℝ) [2, pp. 202-208; 385-391]).
In conclusion, we denote by C r (M, ℝ) the vector space of parallel or covariantly constant r-forms on M with respect to Riemannian symmetric spaces are also a classical 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-Chandra, Helgason, and others developed harmonic analysis and representation theory on these spaces and their Lie groups of isometries (see, for example, [2, pp.235-264]; [13]; [17]; [26]; [38]; [39, pp. 222-292]).
We recall here that the Riemannian symmetric space is a finite dimensional Riemannian manifold   M,g , such that for every its point x there is an involutive geodesic symmetric x s , such that x is an isolated fixed point of x s .  M,g is said to be Riemannian locally symmetric if its geodesic symmetries are in fact isometric.This is equivalent to the vanishing of the covariant derivative of the curvature tensor R of   M,g (see [39, p. 244]).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 socalled 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 type then   M,g is a compact Riemannian manifold with non-negative sectional curvature and positive-definite Ricci tensor (see [39, p. 256]).
Let   M,g be a Riemannian locally symmetric space, then 0   Ric for the Ricci tensor Ric of   M,g .If, in addition,   M,g is irreducible, then it is Einstein (see [2, p. 254]).If in this case, the Einstein constant is positive, then it follows from Myer's theorem (see [2, p. 171]) that the space   M,g is compact.Then one can show that the curvature operator is nonnegative (see also [2, p. 254]).Therefore,   M,g is a Riemannian globally symmetric space of compact type.If, in addition,   M,g is simply-connected and its curvature operator is positivedefinite, 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.
On the other hand, if   M,g is a Riemannian globally symmetric spaces of non-compact type then   M,g is a complete non-compact Riemannian manifold with non-positive sectional curvature and negative-definite Ricci tensor, and diffeomorphic to a Euclidean space (see [39, p. 256]).
We know that an irreducible Riemannian locally symmetric space   M,g is Einstein (see [2, p. 254]).If the Einstein constant is negative, then it follows from Bochner's theorem on Killing fields that the space is noncompact (see [2, p. 254]).In this case, one can show that the curvature operator is nonpositive (see also [2, p. 254]).Therefore,   M,g is a Riemannian globally symmetric space of noncompact type.

CONFORMAL KILLING FORMS ON RIEMANNIAN GLOBALLY SYMMETRIC SPACES
In this section we consider the natural Hilbert space . First, we prove the following theorem for closed and coclosed conformal Killing 2 L -forms on a Riemannian symmetric space of noncompact type that is a complete and simply connected Riemannian manifold of nonpositive sectional curvature.Side by side, its curvature operator R is nonpositive (see [39]).Proof.In our paper [31] we have proved that an arbitrary closed (resp.coclosed) conformal Killing


. If we suppose that the volume is infinite then we obtain a contradiction with our condition that


. It remains to recall that a Riemannian globally symmetric spaces of non-compact type   M,g is a 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 2 L -forms is true.Proof.In our paper [31] we have proved that an arbitrary conformal Killing 2 L -form of degree r is parallel form on r- 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 beautiful account of the rank-one symmetric spaces from 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 spaces using his results.Theorem 3. Let   M,g be a simply connected symmetric space of rank one type.If, in addition,   M,g is a space of compact type, and of odd dimension

 
M,g is a simply connected manifold of odd dimension 1 2   k n then   M,g is a sphere of constant sectional curvature (see [13]).The vector spaces over ℝ of conformal Killing, coclosed and closed conformal Killing p-forms on an Euclidian n-sphere have finite dimensions which are equal to , respectively (see [15] and [16]).This proves our Theorem 3.

1 D
forms and taking their values in the space of homogeneous tensor fields on   g M, .Bourguignon has proved the existence of three basis natural operators of this space, but only the following two 1 D and 2 D of them were recognized.The first operator is the exterior differential operator  as co-closed conformal Killing form.Therefore, the space of all co-closed conformal Killing r-forms we can define as K r (M, ℝ)  T r (M, ℝ)  F r (M, ℝ).A coclosed conformal Killing form is also called as the Killing-Yano tensor (see, for example, [9, pp.426-427]; [10; pp.559-564]).In turnform  as a closed conformal Killing form or closed conformal Killing-Yano tensor (see, for example, for example,[37 pp.202]).

Theorem 1 . 2 L
Let   M,g be an n-dimensional (n  3) simply connected Riemannian symmetric space of noncompact type.Then every closed or co-closed conformal Killing

2 L
-form  is parallel on a complete noncompact Riemannian manifold   M,g with nonpositive curvature operator.It means that C r (M, ℝ)  P r (M, ℝ)  K r (M, ℝ).In this case, const 2 

Theorem 2 .
Let   M,g be a r- symmetric space of noncompact type.Then every conformal Killing 2 L -form of degree r is parallel form on   g M, .If the volume of   g M, is infinite, then every conformal Killing 2 L -form of degree r is identically zero on   g M, .
We consider a rank-one Riemannian symmetric space   M,g of compact type.If, in addition,