On the Killing Tensor Fields on a Compact Riemannian Manifold Grigorios Tsagas Abstract Let (M, g) be a compact Riemannian manifold of dimension n The aim of the present paper is to study the dimension of K q (M, R) in the connection with the Riemannian metric g on M Mathematics Subject Classification: 53C20 Key words: Riemannian manifold, Killing tensor field, Riemannian metric, harmonic q-form and Killing q-form Let (M, g) be a compact Riemannian manifold of dimension n Let K q (M, R) where q = 2,, n be the vector space of Killing tensor fields of order q on M The study of the dimension of K q (M, R) is an important problem This importance comes from the fact there is a connection between q-harmonic forms and Killing tensor fields of order q Let H q (M, R) be the vector space of harmonic q-forms It is known that dim(h q (M, R)) = b q is the q-betti number of M, which is topological invariant It is still open if dim(k q (M, R)) for q = 2,, n is also a topological invariant The aim of the present paper is to study this problem We also improve Yano s results ([6]) The whole paper contains three paragraphs Each of them is analyzed as follows In the second paragraph we study differential operators of cross sections of a fibre bundle over a compact Riemannian manifold M The Killing tensor fields of order q can be considered as special cross sections of the fibre bundle q (T (M)) over M The space of Killing tensor fields K q (M, R) of order q with the connection of the Riemannian metric g on M is studied in the last paragraph These results are an improvement Yano s results ([6]) Balkan Journal of Geometry and Its Applications, Vol, No, 996, pp 9-97 c Balkan Society of Geometers, Geometry Balkan Press
92 GrTsagas 2 Let (M, g) be a compact Riemannian manifold of dimension n without boundary We denote by q (T (M)) and q (T M) the fibre bundles of antisymmetric convariant tensor fields of order q and antisymmetric contravariant tensor fields of order q respectively on the manifold M It is known that the vector space q (T M) coincide with the vector space q (M) of exterior q-forms We must notice that each exterior q-form w is a cross section of q (T M) = q (M) The same is true for each element λ q (T M) The Laplace operator is a second order elliptic differential operator C ( q (M)), that is = dδ δd : C ( q (M)) C ( q (M)), = dδ δd : α (α) = dδ(α) δd(α), α C ( q (M)), where α an exterior q-form and d, δ are first order differential operator defined by d : C ( q (M)) C ( q (M)), δ : C ( q (M)) C ( q (M)) These differential operators are related by < α, δβ >=< dα, β >, α C ( q (M)), β C ( q (M)), where <> is the global inner product on C ( q (M)) The local inner product is defined by < α, γ > = α i,,i q β i,,i q = g j i g j qi q α i,,i q β j,,j q Let (x,, x n ) be a local coordinate system on the chart (U, ϕ) and let {e,, e n } be the associated local frame in M, that is e =,, e n = x x n If α is a q-form, which is a cross section of q (M), that is α C ( q (M)), then α with respect to the local coordinate system can be expressed by α(e i, e i2,, e iq ) = α i,,i q, i < i 2 < < i q n The following formulas are known (2) (dα) i i q j = q! εkj j q i i qj kα j j q j (22) (δα) i2 i q = l α l i 2i q ( α) i i q = k k α i i q
On the Killing Tensor Fields 93 (23) (q )! εkj 2j q i i q ( l k α l i 2 i q k l α l i 2 i q ), where ε iir j j r = if (i i r ) is even permutation of (j j r ) if (i j r ) is odd permutation of (j j r ) 0 if (i i r ) is not permutation of (j j r ) The formula (23), by means of Ricci s formula, becomes l k αi l 2i q k l αi l 2i q = Rrlkα l i r 2i q (24) q Ri r slkαi l 2i s ri si q, s=2 and after some estimates, takes the form ( α) i i q = k k α i i q (q )! εkj 2j q i i q R kl αj l 2 j q (25) 2(q 2)! εklj3jq i i q R klmn αj mn 2 j q If α is a q-form, then we have (26) 2 ( α 2 ) = (α, α) α 2 (q )! L q(α), where q 2 and (27) α 2 = q! k α i i q k α ii q, (28) L q (α) = (q )R klmn α kli 3i q α mn j 3 i q 2R kl α ki 2i q α l i 2 i q From (28) we can consider L q q (M, R), that is as a quadratic form on the vector space (29) L q : q (M, R) R, L q : α L q (α) A q-form α is called killing q-form if its covariant derivative α is a (q )- form This in local system (x,, x n ) can be expressed as follows (20) j α ii2i q i α ji2i q = 0, which is equivalent to (2) q j α ii 2i q i α ji2i q iq α ii 2i q j = 0
94 GrTsagas If α is a killing q-form, then from (2) we obtain (22) j α j i 2 i q = 0 The killing q-form α satisfies the equations q qg jk k j α i i q s α i i s r i s i q R r i s q (22) α ii s r i s i t µ i t i q R rµ i s i t = 0 where s<t Hence if we consider the second order elliptic differential operator D q : C ( q (M, R)) C ( q (M, R)) D q : α D q α, q (D q α) i i q = q g jk k j α i i q s α i i s r i s i q R r i s q (23) s<t Therefore the Ker(D q ) of D q, that is α ii s r i ri t µ i ti q Ker(D q ) = {α q (M, R)/D q (α) = 0} consists of the killing q-forms, whose space is denoted by K q (M, R), that means K q (M, R) = Ker(D q ) Proposition 2 There is an isomorphism between the vector spaces AD q (M, R) and AD q (M, R), where AD q (M, R) and AD q (M, R) are the vector spaces of antisymmetric convariant tensor fields of order q, that is q-forms, and antisymmetric contravariant tensor fields of order q respectively Proof Let (U, φ) be a chart of M with local coordinate system (x,, x n ) If w is a q-form on M, then w has the following components {w ii q / i < i 2 < < i q n}, with respect to the local coordinate system (x,, x n ) We consider the following linear mapping F : AD q (M, R) = q (M, R) AD q (M, R) F : w F (w)
On the Killing Tensor Fields 95 whose component of F (w) with respect to (x,, x n ) are the following F (w) i j q = g i j g i qj q w ii q It can be easily proved that F is bijective Therefore the vector spaces AD q (M, R) and AD q (M, R) are isomorphic, qed Remark 22 If w is a killing q-form, then F (w), which is an antisymmetric contravariant tensor field of order q, has the property F (w) = 0 An antisymmetric contravariant tensor field β of order q with the property β = 0, is called killing tensor field of order q Due to isomorphism F we can use the notion killing tensor field of order q instead of killing q-form and conversely 3 The set of killing tensor fields of order q is denoted by K q (M, R), which is isomorphic onto K q (M, R) In this paragraph we shall study the dim(k q (M, R)) with respect to some properties of the Riemannian metric g on M If α is a killing q-form, then (3) (α, α) ( α) i i q α iiq, which by means of (25) and after some estimates and taking under to consideration (28) we obtain (32) (α, α) = (33) (q ) L q (α) q! The equation (26) by means of (32) becomes 2 ( α 2 ) = α 2 (q ) L q (α) q! From the second order elliptic differential operator D q we obtain and endomorphism (D q ) x of the fibre q (M, R) x in x, that is (34) (D q ) x : q (M, R) x q (M, R) x, which satisfies the relation < (D q ) x u, v >=< u, (D q ) x v >, u, v q (M, R), where <> is the inner product on q (M, R) x induced by the inner product on T M Now, we define (35) R(x) = Sup {< (D q ) x v, v > /v q (M, R), < v, v >= } (36) R max = Sup{R(x)/x M}
96 GrTsagas Now, we shall prove the theorem Theorem 3 Let (M, g) be a compact Riemannian manifold of dimension n If R(x) 0 and there exists an x 0 such that R(x 0 ) < 0, then K q (M, R) = {0} If R max = 0, then dimk q (M, R) = rank{ q (M, R)} Proof If we integrate (33) on the manifold M, we obtain [ (37) α 2 q 2 ] L q (α) dm = 0 2q! M From the inequalities (38) α 2 0 and the assumptions that R(x) 0, imply x M {x 0 } and R(x 0 ) < 0, which (39) L q (x) 0, x M {x 0 } and L q (x 0 ) < 0, we conclude that (30) α = 0 and α/x = 0, x M, which yields α = 0 This proves that K q (M, R) = {0} If R max = 0, then the formula (37) implies (3) [ α 2 ]dm q 2 L q (α)dm 0 q! M which implies α = 0, that means α is a parallel tensor field Hence every killing tensor field of order q on M is parallel Since the maximal number of independent parallel killing tensor fields on M is less or equal than the rank(e), where E is the vector bundle of exterior q-forms, then we have M dim(k q (M, R) = rank{ q (M, R)} qed References [] PHBérard, Spectral Geometry: Direct and Inverse problems Lecture Notes in Mathematics, No 207, Berlin, 986 [2] CBarbance, Transformations conformes d une variété riemanniene, CamptRend 260 (965), 547-549 [3] SBochner, Vector fields and Ricci curvature, Bull Am Math 52 (946), 776-796
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