THE TANGENT BUNDLES and HARMONICITY 1. Dedicated to Professor Gh.Gheorghiev on the occasion of his 90-th birthday.

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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Toul XLIII, s.i.a, Mateatică, 997, f. THE TANGENT BUNDLES and HARMONICITY BY C. ONICIUC Dedicated to Professor Gh.Gheorghiev on the occasion of his 90-th birthday.. Introduction. The probles studied in this paper are concerned with the haronicity of the canonical projection π : T M M, where (M, g is a Rieannian space and T M is its tangent bundle, and conversely, the haronicity of the vector fields ξ χ(m thought of as appings fro M to T M. These things have been studied when T M is endowed with the Rieannian Sasaki etric (see [4] and [8] or when T M is endowed with a pseudo Rieannian etric of coplete lift type g c (see [6] and [7]. In this paper we copute the Levi Civita connection G and its curvature tensor field G K for the Cheeger Grooll etric G. Then we study the haronicity of the natural projection π : (T M, G (M, g and that of the vector fields ξ : (M, g (T M, G. We prove that if ξ : (M, g (T M, G is an isoetric iersion then ξ is haronic. Then we study the subanifold T M of the unit tangent vectors to M in (T M, S and in (T M, G, where S is the Sasaki etric and the haronicity of the projection π : (T M, G (M, g, π : (T M, S (M, g, where π π /T M. For vector fields ξ with ξ it has been established the relation between the haronicity of ξ : (M, g (T M, G and that of ξ : (M, g (T M, S. In the last part of the paper, we study the haronicity of the apping ξ : (M, g (T M,, for ξ, where is the connection obtained Counicated at the Conference Zilele Universităţii Al.I.Cuza, Iaşi, October, 25 30, 996; Învăţăîntului, Universi- Partially supported by the Grant 228/996, Ministerul tatea Al.I.Cuza, Iaşi.

2 52 C. ONICIUC 2 fro the Levi Civita connection of g c by projection on T M with respect to the Liouville vector field on T M, by using Gauss Weingarten type forulas. We study also the haronicity of the etrics g c, S and G, each with respect to the others. The author wishes to express his gratitude to professor V. Oproiu for any helpful talks and hints concerning the arguent discussed in this paper. 2. The tangent bundles and Rieannian etrics. Consider an n diensional, sooth, Rieannian anifold (M, g and let π : T M M be the tangent bundle of M. Then T M is a 2n diensional anifold and on T M we ay use soe special local charts induced fro local charts on M. If (U, x i ; i,..., n is a local chart on M then the local chart (π (U, x i, y i is induced on T M where x i x i π and y i are the vector space coordonates of v π (U T M with respect to the natural local frae ( x,..., ( x i.e. v y i n x π(v. i Denote by the Levi Civita connection of g. The connection deterines a horizontal distribution HT M on T M and the local vector fields: ( x i x i yj Γ h ij y h ; i,..., n define a local frae in HT M where Γ k ij are the Christoffel sybols. We have T T M HT M V T M where V T M Ker π is the vertical distribution on T M and the local vector fields y ; i,..., n define a local frae i of V T M. Then we ay define the horizontal lift X H of a vector field X ξ i χ(u by: x i and the vertical lift of X by: X H ξ i x i χ(π (U X V ξ i y i χ(π (U. It follows ( x H i x, ( i x V i y (see [9]. i The syste of local fore (dx ( i, y i ; i,..., n on T M defines the local dual frae of the local frae ; i,..., n on T M where: x, i y i (2 y i (dx i H dy i + y j Γ i hjdx h.

3 3 THE TANGENT BUNDLES AND HARMONICITY 53 We ay consider several etrics on T M by using the above horizontal and vertical lifts. The ost known is Sasaki etric, S, defined by: (3 S(X H, Y H g(x, Y, S(X V, Y V g(x, Y, S(X H, Y V 0. S is given locally by: (4 S g ij dx i dx j + g ij y i y j. ( gij 0 It follows that the atrix associated with S is:. 0 g ij The Sasaki etric is a Rieannian etric. The horizontal and vertical distributions are orthogonal with respect to S, and the fibres of horizontal and vertical distributions are canonically isoetric with the tangent spaces on M in the corresponding points. The Cheeger Grooll etric, G, is a Rieannian etric defined by: (5 { G(X V, Y V v + v 2 {g p (X, Y + g p (X, vg p (Y, v} G(X H, Y V 0, G(X H, Y H g(x, Y where v T M, v is the nor of v T p M with respect to g p, p π(v (see [5]. Locally, G is given by: (6 G g ij dx i dx j + + g 00 {g ij + g i0 g j0 }y i y j where g i0 g ik y k and g 00 g i0 y i. The atrix associated with G is ( gij 0 0 +g 00 (g ij + g i0 g j0 The horizontal and vertical distributions are orthogonal each other with respect to G and the fibres of the horizontal distribution are isoetric with the tangent space on M in the corresponding points. The coplete lift type etric g c is a pseudo Rieannian etric on T M defined by: (7 g c (X H, Y H 0, g c (X V, Y V 0, g c (X V, Y H g(x, Y g c is written locally by: (8 g c 2g ij y i dx j ;

4 54 C. ONICIUC 4 ( 0 gij its atrix being: and having the signature (n,n. We can see g ij 0 that the horizontal and vertical distributions are axially isotropic and g c defines a pairing between the. A slight generalization of the coplete lift consists in: (9 g c (X H, Y H c(x, Y, g c (X V, Y H g(x, Y, g c (X V, Y V 0, where c is a syetric tensor field of type (0, 2 on M. In this case, the horizontal distribution is no longer isotropic. Locally, if c c ij dx i dx j, then: (0 g c c ij dx i dx j + 2g ij y i dx j. The Levi Civita connections S and of S and g c etrics respectively have been calculated so far. We also have calculated their curvature tensors s K and K respectively (see [8],[6]. We ay prove by a straightforward coputation the following results: by: ( Proposition. The Levi Civita connection G of G is given locally G y i G y i G x i G x i y j +g 00 {g ij C g i0 A l ij y l x j y j 2(+g 00 Rh ji0 x h y g j j0 y }+ i 2(+g 00 Rh ij0 + Γ h x h ij, y h + Γ h ij x h x j 2 Rh 0ij y h (+g 00 2 {g ij +g i0 g j0 }C where C y i y i is Liouville vector field on T M and A l ij { (2g ij y l + g ig l j0 l jg i0 00 ( + g 00 2 (g ijg 00 y l g j0 g i0 y l } where Rijk h are the local coordinate coponents of the curvature tensor field, R of. Fro (0 it follows easily the global expression of G. By a straightforward coputation we also get the following results:

5 5 THE TANGENT BUNDLES AND HARMONICITY 55 by: (2 Proposition 2. The curvature tensor field G K of G is locally given ( G K x i, { + Rkij l + x j x k 2 ( krijy l y l + } 4( + g 00 (Rh 0jkRih0 l R0ikR h jh0 l 2R0ijR h kh0 l x l ( G K x i, { + Rkij l G K ( y i, x j y k 2( + g 00 ( jrik l i Rjky l + g 00 g k0 R l 0ij + + ( + g 00 2 R k0ij(2 + g 00 C y j x k + g 00 G K ( y i, 4( + g 00 (Rh jk0r l 0ih R h ik0r l 0jh x l + } y l + { [ Rkij l + (g i0 Rkj0 l g j0 R + g ki0+ l 00 + ]} 4 (Rh kj0rhi0 l Rki0R h hj0 l x l x j y k 2( + g 00 Rh jki x h + + 2( + g 00 2 (g i0rjk0 h g k0 Rji0 h x h + 4( + g 00 2 Rl jk0rli0 h x h x j x k 2( + g 00 ( jrkiy l x l + ( G K y i, { + 2 Rl ijk + 2( + g 00 g i0r l 0jk 4( + g 00 Rh ki0r l 0jh } y l 2( + g 00 2 R i0jk(2 + g 00 C ( { } G K y i, y j y k 3 g jk + g 00 y i g ik y j + { ( } + ( + g (g ik g j0 g jk g i0 C + 2 g ik y j g jk y i g 00 + { + ( + g (g i0 g jk g j0 g ik ( + 2g 00 C+ ( ( } + g ik y j g jk y i g 00 + g k0 g 00 g i0 y j g j0 y i

6 56 C. ONICIUC 6 Fro (2 we can easily get the global expression of G K. Reark. G is flat if and only if is flat. Fro (2 we have: G K v (X H, Y H C + v 2 {R(X, Y v}v, G K v (C, X V Y H ( + v 2 {R(v, XY }H 2 G K v (X H, Y H C 2( + v 2 2 {R(v, XY }H, G K(C, X H C 0, G K(C, SC 0 G K(C, SS 0, G K v (C, X H Y H {R(X, Y v}v 2( + v 2 where S y i x is the geodesic spray defined by the connection. i Let M and N be two Rieannian anifolds and let f : M N be a sooth ap. The local expression of the second fundaental for of f denoted by β(f is (3 β(f( x i, x j {f ij α M Γ k ijfk α + N Γ α βγf β i f γ j } u α and that of the tension field τ(f of f is: (4 τ(f g ij {f α ij M Γ k ijf α k + N Γ α βγf β i f γ j } u α (see [2]. Fro (3 and (4 we reark that β(f and τ(f ay also be defined when N is only endowed with a linear connection without torsion or when M is only a pseudo Rieannian anifold. Thus is has sense to study the haronicity of the vector fields regarded as aps ξ : M T M, ξ : M T M, when we no longer consider on T M, T M, the Levi Civita connection of a Rieannian (pseudo Rieannian etric but another connection, without torsion, obtained fro the Levi Civita connection by various ethods. It also will have sense the study of the haronicity of the canonical projection π : T M M, when we have a pseudo Rieannian etric on T M.

7 7 THE TANGENT BUNDLES AND HARMONICITY 57 Now, we reark that the canonical projection π : (T M, G (M, g is a Rieannian subersion. The we get by a straightforward coputation: (5 ( β(π ( β(π v y, i y j y, i x j β(π ( x i, x j 0 2(+ v 2 yl R h jil (π(v x h Fro (5 and using results fro the theory of the Rieannian subersions (see [3] we obtain: Theore. The Rieannian subersion π : (T M, G (M, g has the following properties: a π has its fibres totally geodesic, hence inial b π is haronic c the following relations are equivalent: π is totally geodesic the horizontal distribution is integrable is flat. Let c be a Rieannian etric on M. We consider π : (T M, G (M, c. We have (6 { β(π ( y, i y j 0, β(π v ( y, i x j β(π ( x i, x j { c Γ h ij Γh ij } x h 2(+ v 2 yl R h jil (π(v x h where c Γ h ij are the Christoffel sybols of the etric c. So we obtain: Proposition 3. π : (T M, G (M, c is totally geodesic if and only if (M, g is flat and : (M, g (M, c is totally geodesic. We reark that if π : (T M, G (M, c is totally geodesic then (M, g and (M, c are flat. Let g and c be two Rieannian etrics on M. We say that c is haronic with respect to g if (7 g ij { c Γ h ij Γ h ij} 0 (see []. We observe that this definition can also be extended for the pseudo Rieannian case. Fro (6 it follows that:

8 58 C. ONICIUC 8 Proposition 4. π : (T M, G (M, c is haronic if and only if c is haronic with respect to g. Consider the tensor fields T and A on T M given by: (8 T ( X, Ỹ HG V X V Ỹ + V G V X HỸ, A( X, Ỹ HG H X V Ỹ + +V G H X HỸ failiar in the theory of Rieannian subersions. Reark that T is called O Neill s tensor. Since π : (T M, G (M, g is a Rieannian subersion with its fibres totally geodesic, or by straightforward coputation, we obtain that T 0. For A we obtain the following expression: (9 ( A So we get: y, i ( y A j y, i x 0 j ( A ( x i, x j 2 Rh 0ij y h, A x, i y j 2(+g 00 Rh ij0 x h Proposition 5. The following relations are equivalent: a A 0 b the horizontal distribution is integrable c A(X H, Y H 0, X, Y χ(m d A(X H, Y V 0, X, Y χ(m Reark. Fro theore and proposition 5 it results that π : (T M, G (M, g is totally geodesic if and only if A 0. The Schouten connection G associated with G is defined by: (20 G X Ỹ V G X V Ỹ + HG X HỸ, X, Ỹ χ(t M Locally G is given by: (2 { G y i G x i y j G y i y j, G y i y j Γ h ij y h, G x i x j x j Γ h ij G y i x h x j The ean connection of the connection G is G G 2 T where T is the torsion tensor field of G. In local coordinate G has the expression: G y (22 i y G j y i y, G j y i x G j 2 y i x j G x i y G j x i y + j 4(+g 00 Rh ij0, G x h x i x G j x i x j

9 9 THE TANGENT BUNDLES AND HARMONICITY 59 Obviously G is without torsion. We reark that G G if and only if R 0. Let : (T M, G (T M, G be the identity ap. We obtain: ( ( ( (23 β( y i, y j β( x i, x j 0, β( y i, x j 4( + g 00 Rh ji0 x h hence τ( trace β( 0. Obviously, if R 0 then β( 0. Now, let ξ be a vector field. We consider ξ as a sooth ap fro M to T M, ξ : (M, g (T M, G. Proposition 6. ξ : (M, g (T M, G is an isoetric iersion if and only if ξ is parallel i.e. ξ 0. Proof. We obtain iediatly: (24 ξ,p X (X H + ( X ξ V ξ(p, p M If we consider g p (X, Y G ξ(p (ξ,p X, ξ,p Y g p (X, Y + + ξ(p 2 {g p( X ξ, Y ξ+ +g p ( X ξ, ξg p ( Y ξ, ξ} then ξ is isoetric iersion if and only if g g. We reark that g g if and only if X ξ 0 for all X χ(m i.e. ξ 0. Considering the relations (24 and ( we obtain by a straightforward coputation: Proposition 7. The ap ξ : (M, g (T M, G has β(ξ and τ(ξ given by: (25 β(ξ ( x i, x j { 2 Rh ijξ + i j ξ h + ( i ξ l ( j ξ k A h lk} y h 2( + ξ 2 {( jξ k Rikξ h + ( i ξ l Rjlξ h } x h (26 τ(ξ {g ij ( i j ξ h + g ij ( i ξ l ( j ξ k A h lk} y h + ξ 2 {gij ( j ξ k Rikξ h } x h In a siilar way, fro the relations (24 and (22 we obtain:

10 60 C. ONICIUC 0 Proposition 8. The ap ξ : (M, g (T M, G has the second fundaental for, β(ξ, and the tension field τ (ξ given by: (27 ( β(ξ x i, x j { 2 Rh ijξ + i j ξ h + ( i ξ l ( j ξ k A h lk} y h 4( + ξ 2 {( jξ k R h ikξ + ( i ξ l R h jlξ } x h (28 τ (ξ {g ij ( i j ξ h + g ij ( i ξ l ( j ξ k A h lk} y h 2( + ξ 2 {gij ( j ξ k Rikξ h } x h Coparing the relations (25 and (26 to (27 and (28, we obtain: Theore 2. β(ξ 0 if and only if β(ξ 0 and τ(ξ 0 if and only if τ (ξ 0. So, the use of the connection G has not led us to new results. Fro (26 it follows that: Theore 3. ξ : (M, g (T M, G is haronic if and only if it verifies the following syste: (29 g ij ( i j ξ h + g ij ( i ξ l ( j ξ k A h lk 0, g ij ( j ξ k R h ikξ 0 Fro the proposition 6 and (25 it follows that: Theore 4. If ξ : (M, g (T M, G is isoetric iersion then ξ is totally geodesic, hence ξ is haronic. We note that ξ is parallel if and only if ξ is an isoetric inial iersion. Now we shall study the relations (25 and (26 becoe in soe particular cases: Suppose that ξ. Then β(ξ and τ(ξ have the following expressions: β(ξ ( x i, x j { 2 Rh ijξ + i jξ h ( iξ l g lk ( j ξ k ξ h } y h 4 {( jξ k R h ikξ + ( i ξ l R h jlξ } x h

11 THE TANGENT BUNDLES AND HARMONICITY 6 τ(ξ {g ij ( i j ξ h gij ( i ξ l g lk ( j ξ k ξ h } y h 2 {gij ( j ξ k R h ikξ } x h 2 Suppose that ξ and (M, g has its sectional curvature constant a. Then: ( β(ξ x i, x j { a 2 (ξ ji h ξ i j h + i j ξ h ( iξ l g lk ( j ξ k ξ h } y h a 4 {( jξ h ξ i ξ h ( j ξ i + ( i ξ h ξ j ξ h ( i ξ j } x h τ(ξ {g ij ( i j ξ h gij ( i ξ l g lk ( j ξ k ξ h } y h a 2 {gij [( j ξ h ξ i ξ h ( j ξ i ]} x h 3 Suppose that ξ, ξ Killing and (M, g has sectional curvature constant a. Then: ( β(ξ x i, x j a 2 {ξ ji h + ξ i j h 2 g ijξ h 3 2 (ξ iξ j ξ h } y h a 4 {( jξ h ξ i + ( i ξ h ξ j } x h τ(ξ a ( nξh 4 y h So if a 0 then ξ is totally geodesic, and if a 0 then ξ is not haronic. 4 Suppose that ξ, ξ 0 i.e. g ij ( i j ξ h R h l ξl where R jk is the local coordinate expression of the Ricci tensor field ρ(r, R i k R jkg ij and (M, g has sectional curvature constant a. Then: τ(ξ a ( nξh 4 y h a 2 {gij [( j ξ h ξ i ξ h ( j ξ i ]} x h So ξ is haronic if and only if a 0. 5 Suppose that ξ. Then τ(ξ is colinear with C y i y if and only i if: g ij ( i j ξ h fξ h, g ij ( j ξ k ξ Rik h 0

12 62 C. ONICIUC 2 where f F(M. 6 Suppose that ξ, (M, g has its sectional curvature constant a, ξ 0 and we also suppose that g ij [( j ξ h ξ i ξ h ( j ξ i ] 0. Then τ(ξ is colinear with C. Now we return to Sasaki etric S. It is known that π : (T M, S (M, g is a Rieannian haronic subersion with totally geodesic fibres. Let c be a Rieannian etric on M. One can prove easily: Proposition 9. The etric c is haronic with respect to g if and only if π : (T M, S (M, c is haronic. The ean connection S of the Schouten connection associated with S has the local coordinate expression: (30 S y i S x i y j 0, S y i x j 4 Rh ji0 x h y Γ h j ij y h 4 Rh ij0, S x h x i x j Γ h ij x h 2 Rh 0ij y h We reark that S S if and only if R 0. Let : (T M, S (T M, S be the identity ap. We obtain by a straightforward coputation τ( trace β( 0; evidently, if R 0 then β( 0. Now let ξ be a vector field. We consider ξ as a ap fro M to T M. Proposition 0. a ξ : (M, g (T M, S has its tension field τ(ξ given locally by: (3 τ(ξ {g ij ( i j ξ h } y h {( jξ l R h ilξ g ij } x h b ξ : (M, g (T M, S has the tension field τ (ξ given locally by: (32 τ (ξ {g ij ( i j ξ h } y h 2 {( jξ l R h ilξ g ij } x h. Fro (3 and (32 it follows:

13 3 THE TANGENT BUNDLES AND HARMONICITY 63 Theore 5. For ξ χ(m we have τ(ξ 0 if and only if τ (ξ 0. Rearks. Fro (30 it follows that ξ : (M, g (T M, S is haronic if and only if ξ verifies the equations syste: g ij ( i j ξ h 0, g ij ( j ξ k ξ R h ik 0 2 If ξ is a Killing vector field or a geodesic vector field or ξ 0 and (M, g is flat, then τ(ξ 0, i.e. ξ is haronic. 3 If ξ is a Killing vector field and (M, g has the sectional curvature constant a, a 0, then: τ(ξ a( nξ h y h a{gij ( j ξ h ξ i } x h Hence ξ is not haronic. 4 If ξ 0 and (M, g has the sectional curvature constant a, a 0, then: τ(ξ a( nξ h y h a{gij [( j ξ h ξ i ξ h ( j ξ i ]} x h. Hence ξ is not haronic. 5 If M is copact and orientable, then ξ is haronic if and only if ξ The subanifold T M of T M. Let T M {v T M; v } be the 2n diensional subanifold of T M. The local vector fields {Y i, x } n i i generate locally T T M, where (33 Y i y i g i0c. We consider T M as a subanifold of (T M, G. We verify easily that C y i y is orthogonal on T i M with respect to G. We get by a straightforward coputation: Proposition. We have the following relations: B ( ( x, i x B j x, Y i j 0, B(Yi, Y j 4 (g i0g j0 g ij C ( (34 H ( C x, i x j HC x, Y i j 0, HC (Y i, Y j 4 (g i0g j0 g ij, ( S C x 0, i SC (Y i 2 Y i where B is the second fundaental for of subanifold T M of the anifold (T M, G and: H C (X, Y G(B(X, Y, C, X, Y χ(t M S C (X ( G X C T We reark that T M is not totally geodesic in (T M, G, it is not inial and it is not totally ubilical. Since T M is not inial it follows that i : (T M, G /T M (T M, G is not haronic.

14 64 C. ONICIUC 4 Proposition 2. The Levi Civita connection G of the Rieannian etric G /T M is given locally by: (35 { G Y j x i 4 Rh ij0 x h G Yi x j 4 Rh ji0 + Γ h ij Y h, G x i x j x h, G Yi Y j g j0 Y i 2 Rh 0ij Y h + Γ h ij x h We consider now T M as a subanifold of (T M, S; in the sae way we obtain that C is orthogonal on T M with respect to S and: Proposition 3. We have: B ( x, i x B( j x, Y i j 0, B(Y i, Y j (g i0 g j0 g ij C, ( (36 H C x, i x j HC ( x, Y i j 0, H C (Y i, Y j g i0 g j0 g ij, ( S C x 0, i SC (Y i Y i Hence we get that T M is not totally geodesic, is not inial and is not totally ubilical in (T M, S; i : (T M, S /T M (T M, S is not haronic. Proposition 4. The Levi Civita connection S of the Rieannian etric S /T M is given locally by: (37 { S Y j x i 2 Rh ij0 x h S Yi x j 2 Rh ji0 + Γ h ij Y h, S x i x j x h, S Yi Y j g j0 Y i 2 Rh 0ij Y h + Γ h ij x h Using the Gauss Weingarten relations we project S on T M with respect to the vector field C and we get S G. Proposition 5. a T M is a hypersurface of the (T M, S having the ean curvature constant: (38 H T M S n 2n C b T M is a hypersurface of the (T M, G having the ean curvature constant: (39 H T M G n 2(2n C Hence it follows that i : (T M, G /T M (T M, G is neither inial nor pseudo-ubilical. Siilarly for i : (T M, S /T M (T M, S. Now studying the haronicity of the etrics G and S each with respect to other, fro the expressions of G and S, we obtain:

15 5 THE TANGENT BUNDLES AND HARMONICITY 65 Proposition 6. Let T M : (T M, G (T M, S be the identity ap. Then: a T M cannot be totally geodesic b T M has the tension field τ G ( T M given by: (40 τ G ( T M (n (g C, g 00 + so τ G ( T M is orthogonal on T M and S cannot be haronic with respect to G. Proposition 7. Let T M : (T M, G (T M, S be the identity ap. We have: a T M is totally geodesic if and only if R 0 b T M is haronic i.e. S /T M is haronic with respect to G /T M. Proposition 8. Let T M : (T M, S (T M, G be the identity ap. Then T M has the tension field τ G ( T M given by: (4 τ S ( T M (n (g (g C Proposition 9. Let T M : (T M, S (T M, G be the identity ap. Then T M is haronic, so G /T M is haronic with respect to S /T M. Let π be defined by : π π /T M. We observ that π : (T M, G (M, g is a Rieannian subersion. Fro the relations: ( (42 π (Y i 0, π x i x i, and relations (35, we obtain: Theore 6. a π : (T M, G (M, g is totally geodesic if and only if R 0 b π : (T M, G (M, g is haronic. In the sae way, for π : (T M, S (M, g we get: Theore 7. a π : (T M, S (M, g is totally geodesic if and only if R 0 b π : (T M, S (M, g is haronic.

16 66 C. ONICIUC 6 We reark that T M is orientable and if M is copact then T M is copact. So, if M is copact, π is haronic with the eaning of its basic definition (see [2]. Let ξ be a vector field with ξ. Denote by: ξ G : (M, g (T M, G, ξ G (p ξ(p ξ G : (M, g (T M, G, ξ G (p ξ(p, p M; ξ S : (M, g (T M, S, ξ S (p ξ(p ξ S : (M, g (T M, S, ξ S (p ξ(p, p M. We have: ( (43 ξ x i x i + ( iξ j Y j Fro the relations (43, (35 and (37 we obtain: τ(ξ G {g ij ( i j ξ h }Y h 2 {( jξ l Ril h ξ g ij }, x h (44 τ(ξ G τ(ξ G g ij B( x + ( i i ξ l Y l, x + ( j j ξ l Y l, τ(ξ S {g ij ( i j ξ h }Y h {( j ξ l Ril h ξ g ij }. x h Proposition 20. τ(ξ G 0 if and only if τ(ξ G is colinear with C. The conditions under which τ(ξ G is colinear with C have previously been studied. Theore 8. Let ξ be a vector field with ξ. Then: a τ(ξ G 0 if and only if τ(ξ S 0, b τ(ξ G 0 if and only if τ(ξ S 0. Proof. a It results fro the relations (43. b It results fro the relations (26, (3 and fro the relations ξ h ξ h, ( i j ξ h ξ h ( j ξ t g tr ( i ξ r. Reark. If M is copact and orientable, ξ, then τ(ξ G 0 if and only if ξ 0. 2 τ(ξ S 0 if and only if τ(ξ S 0 is equivalent with det( h ξ hξ 0. But det( h ξ hξ 0 because ( h ξ hξ ξ 0 and ξ 0. So it has sense to consider the case ξ. 4. The tangent bundles and the pseudo Rieannian etric g c. We consider the pseudo Rieannian etric g c on T M. Its Levi Civita connection,, has been calculated (see [5]. We consider the canonical projection π : (T M, g c (M, g. We have: ( ( ( (45 β(π y i, y j β(π x i, x j β(π y i, x j 0 Hence

17 7 THE TANGENT BUNDLES AND HARMONICITY 67 Theore 9. π : (T M, g c (M, g has the properties: a β(π 0 b τ(π 0 i.e. π is haronic. Let g be a Rieannian etric on M. (M, g. We have: ( (46 β(π y i, ( y j β(π y i, We consider π : (T M, g c x j 0, ( β(π x i, x j ( Γ k ij Γ k ij x k, where Γ k ij are the Christoffel sybols of g etric. Proposition 2. a π : (T M, g c (M, g is totally geodesic if and only if : (M, g (M, g is totally geodesic; b π : (T M, g c (M, g is haronic. Let c be a syetric tensor field of type (0, 2 on M, and let g c be the pseudo Rieannian etric defined with respect to g and c. We reark that: (47 x i x i ( Γ h 0i y h x i ( Γ h 0i Γ h 0i y h l Proposition 22. g c is haronic with respect to g c if and only if g is haronic with respect to g. We reark that the haronicity of g c with respect to g c does not depend on c or c. Studying the haronicity of the etrics g c, S, G each with respect to other, we obtain: Proposition 23. a Let : (T M, g c (T M, S be the identity ap. Then τ( R0 h x h

18 68 C. ONICIUC 8 so that S is haronic with respect to g c if and only if ρ(r 0. b We consider : (T M, g c (T M, G. We have: τ( (n (g ( + g 00 2 C + + g 00 R h x h hence G is not haronic with respect to g c. c We consider : (T M, S (T M, g c. We have: τ( R0 h x so that τ( 0 if and only if ρ(r 0. h d We consider : (T M, G (T M, g c. We obtain: τ( R h 0 y h (n (g C + g 00 hence g c is not haronic with respect to G. The ean connection of the Schouten connection associated with is locally given by y (48 i y 0, j y i x 0, j y Γ k j ij, y k x Γ h j ij x h 2 Rh 0ij y h x i x i We see that does not depend on c. Proposition 24. We consider : (T M, g c (T M,. We have: a β( 0 if and only if c 0 and R 0. b τ( 0. Let ξ be a vector field. We consider ξ : (M, g (T M,. By a straightforward coputation we get: ( (49 β(ξ x i, x j { 2 Rh ijξ + i j ξ h } y h (50 τ (ξ {g ij ( i j ξ h } y h For ξ : (M, g (T M, g c we have (see [6]: (5 τ(ξ {g ij ( i j ξ h + ξ l R h l + g kh ( i c i k 2 k(tr c} y h

19 9 THE TANGENT BUNDLES AND HARMONICITY 69 Reark. The use of the connection has led us to new results. 2 If ξ is a Killing vector field, or a geodesic one, or ξ 0 and (M, g is flat or (M, g is Ricci flat, then τ (ξ 0 i.e. ξ is haronic. 3 Suppose that ξ is Killing and (M, g has the sectional curvature constant a 0. Then τ (ξ a( nξ h a( nξv yh Hence ξ is not haronic. 4 Assue that ξ is Killing or geodesic and n 2. Then: τ (ξ kξ V, where k F(M; so τ (ξ 0. 5 Assue that ξ 0 and (M, g has the sectional curvature constant a 0. Then τ (ξ a(n ξ V Hence ξ is not haronic. 6 Suppose that ξ 0 and n 2. Then: τ (ξ kξ V, k F(M; 7 If M is copact and orientable, then ξ is haronic if and only if ξ The subanifold T M of (T M, g c. Using the Gauss Weingarten relations we project on T M with respect to the vector field C and we get given locally by: (52 Yi Y j g j0 Y i, Yi x j 0, x i x j x i Y j Γ h ij Y h Fro the relations (43 and (52 we obtain: Γ h ij x h 2 Rh 0ij Y h Proposition 25. Let ξ be a vector field with ξ. We consider ξ : (M, g (T M,. Then β(ξ and τ (ξ are locally given by: (53 β(ξ { 2 Rh ijξ + i j ξ h }Y h (54 τ (ξ g ij ( i j ξ h Y h

20 70 C. ONICIUC 20 Theore 0. τ (ξ 0 if and only if τ (ξ is colinear with C. Reark. if ξ, ξ is Killing or ξ 0 and (M, g has the sectional curvature constant then τ (ξ 0. 2 if ξ, ξ is Killing or ξ 0 and n 2 then τ (ξ 0. In what follows we shall consider c g. In this case the Levi Civita connection becoes (55 y i x i y j 0, y i x j 0, y j Γ k ij y k, x i x j Γ h ij x h + R h j0i y h Consider P S C, S y i x and observe that P spann {Y i i, x }, i g c (P, Y i g ( c P, x 0, g c (P, P. i Projecting on T M with respect to P and we obtain the torsion free connection, given locally by: (56 Yi Y j (g i0 g j0 g ij S g j0 Y i, Yi x 0, j Y j Γ h x i ij Y h, x i x {Γ h j ij + R 0j0iy h } + R h x h j0i Y h Fro the relations (43 and (56 we get: Proposition 26. Let ξ be a vector field with ξ. We consider ξ : (M, g (T M,. expressions given by: (57 Then β(ξ and τ (ξ have the local coordinate ( β(ξ x i, x j {R jli ξ ξ l ( i ξ l g lk ( j ξ k }ξ h x h + +{R h jiξ + i j ξ h }Y h (58 τ (ξ {Rl ξ ξ l g ij ( i ξ l g lk ( j ξ k }ξ h x h + +{R h ξ + g ij ( i j ξ h }Y h Fro (5, (58 we obtain:

21 2 THE TANGENT BUNDLES AND HARMONICITY 7 (59 Theore. The following relations are equivalent: a τ (ξ 0 b τ(ξ 0 c ξ and ξ is a geodesic vector field. Reark. If ξ and ξ Killing then τ (ξ 0. We project now, on T M with respect to C and we get, given by: { Yi Y j g j0 Y i, Yi x j x i x j 0, x i Y j Γ h ij Y h Γ h ij + R h x h j0i Y h, Proposition 27. Let ξ be a vector field with ξ. We consider ξ : (M, g (T M,. The β(ξ and τ(ξ have the local coordinate expression given by: (60 β(ξ ( x i, x j {Rjiξ h + i j ξ h }Y h (6 τ(ξ {R h ξ + g ij ( i j ξ h }Y h Theore 2. τ(ξ 0 if and only if τ(ξ is colinear with C. Reark. if τ (ξ 0 then τ(ξ 0. 2 if ξ, ξ 0 and (M, g has been the sectional curvature constant or n 2 then τ(ξ 0. (62 We project on T M with respect to P. We get: YiY j (g i0g j0 g ijs g j0y i, x i Y j Γ h ij Y h, x i x j Γ h ij Yi x h x j 0 2 Rh 0ij Y h Proposition 28. For ξ with ξ we have: ( β(ξ (63 x i, x j { 2 Rh ijξ + i j ξ h }Y h {( i ξ l g lk ( j ξ k }ξ h x h (64 τ (ξ {g ij ( i j ξ h }Y h {g ij ( i ξ l g lk ( j ξ k }ξ h x h

22 72 C. ONICIUC 22 Theore 3. τ (ξ 0 if and only if τ (ξ 0. Reark. if τ (ξ 0, then τ (ξ 0. 2 if ξ 0 or ξ is Killing vector field and ρ(r 0 or R 0, then τ (ξ 0. 3 if M is copact and orientable then τ (ξ 0 if and only if ξ 0. It is ore convenient to consider the projection on T M using the Liouville vector field C, because the use of P iposes very restrictive conditions for the haronicity of the vector fields. REFERENCES. C h e n, B.Y. and N a g a n o, T. Haronic etrics, haronic tensors and Gauss aps, J.Math. Soc. Japan 36(2 (984, E e l l s, J. and L e a i r e, L. Selected topics in haronic aps, Conf. Board of the Math. Sci. A.M.S. 50(983, E e l l s, J. and R a t t o, A. Haronic aps and inial iersions with syetries. Method of ordinary differential equations applied to elliptic variational probles, Ann. Math. Studies 30, Princeton University Press, I s h i h a r a, S. Haronic sections of tangent bundles, J. Math. Tokushia Univ., 3, 979, M u s s o, E. and T r i c e r r i, F. Rieannian etrics on tangent bundles, Ann.Mat.Pura Appl. (4 50 (988, O n i c i u c, C. On the haronic sections of tangent bundles, in print. 7. O p r o i u, V. On the haronic sections of cotangent bundles, Rend. Se. Fac. Sci. Univ. Cagliari, 59(2, (989, P i u, M.P. Capi di vettori ed applicazioni aroniche, Rend. Se. Fac. Sci. Univ.Cagliari, 52(, (982, Y a n o, K. and I s h i h a r a, S. Tangent and Cotangent Bundle, M.Dekker, New York, 973. Received: 6.XII.996 Faculty of Matheatics, University Al.I.Cuza, 6600 Iaşi, ROMANIA