Some examples of harmonic maps for g-natural metrics

We produce new examples of harmonic maps, having as source manifold a space (M, g) of constant curvature and as target manifold its tangent bundle TM , equipped with a suitable Riemannian g-natural metric. In particular, we determine a family of Riemannian g-natural metrics G on TS2, with respect to which all conformal gradient vector fields define harmonic maps from S2 into (TS2, G).


Introduction
In the theory of harmonic maps, a fundamental question concerns the existence of harmonic maps between two given Riemannian manifolds (M, g) and (M , g ).
If (M, g) is compact and (M , g ) is of non-positive sectional curvature, then there exists a harmonic map f : (M, g) → (M , g ) in each homotopy class [8].However, there is no general existence result when (M , g ) admits some positive sectional curvatures.This fact makes it interesting to find examples of harmonic maps having such a target manifold.Clearly, the non-applicability of the standard existence theory for harmonic maps, leads to the construction of examples by an ad hoc approach.Some examples in this direction are given in [5], where harmonic maps are defined by tangent vector fields.Moreover, given an arbitrary section σ of a Riemannian manifold (M, g), Oniciuc [12] showed how to construct a suitable metric on the tangent bundle T M , which depends on σ and makes σ : (M, g) → T M totally geodesic, hence harmonic.
The Sasaki metric g s is by far the most investigated among all possible Riemannnian metrics on T M .Nouhaud [11] and Ishihara [9] independently proved that parallel vector fields on a compact Riemannian manifold are the only defining harmonic maps V : (M, g) → (T M, g s ).This rigid behaviour of the Sasaki metric with respect to harmonicity, and the fact that g s and other well-known Riemannian metrics on T M are gnatural, motivated in [1] the study of the harmonicity of V : (M, g) → (T M, G), with T M equipped with an arbitrary Riemannian g-natural metric.Riemannian g-natural metrics form a large family of Riemannian metrics on T M , which depends on six independent smooth functions from R + to R. Several different behaviours have been found for the harmonic maps of (T M, G), with respect to different Riemannian g-natural metrics G.In particular, it was proved that, for a suitable choice of G, some remarkable non-parallel vector fields, such as Hopf vector fields on the unit sphere S 2m+1 and, more generally, the Reeb vector field of a contact metric manifold, are harmonic maps of (T M, G).
In this paper, we give new examples of harmonic maps from a Riemannian manifold (M, g) of constant sectional curvature and its tangent bundle T M , equipped with some Riemannian g-natural metric G.More precisely, in Section 4 we show that there exists a family of Riemannian g-natural metrics G on T S 2 (also including some metrics conformal to the Sasaki metric g s ), with respect to which all conformal gradient vector fields define harmonic maps from S 2 into (T S 2 , G).It is worthwhile to remark that (T S 2 , G) necessarily admits some positive sectional curvatures, and that the metrics of Cheeger-Gromoll type h p,q , studied by M. Benyounes, E. Loubeau and C.M. Wood ( [5], [6]), fail to provide examples of harmonic maps from (S 2 , g) to (T S 2 , h p,q ), for any Riemannian metric g on S 2 .In Section 5, we show that if (M, g) has constant non-positive sectional curvature, then a Riemannian g-natural metric can be found, such that all vector fields of a fixed constant length are harmonic maps.In Section 6, we suppose (M, g) flat and characterize harmonic maps for a family of Riemannian g-natural metrics on T M .Within this family, there exist Riemannian g-natural metrics for which the target manifold T M has nonnegative sectional curvature.The definition and basic properties of gnatural metrics on T M will be recalled in Section 2, while the expression of the tension field associated to V : (M, g) → (T M, G) is in Section 3.

Basic formulas on g-natural metrics on tangent bundles
Let (M, g) be an n-dimensional Riemannian manifold and ∇ its Levi-Civita connection.At any point (x, u) of its tangent bundle T M , the tangent space of T M splits into the horizontal and vertical subspaces with respect to ∇: (T M ) (x,u) = H (x,u) ⊕ V (x,u) .For any vector X ∈ M x , there exists a unique vector for all functions f on M .Here we consider 1-forms df on M as functions on T M (i.e., (df )(x, u) = uf ).The map X → X h is an isomorphism between the vector spaces M x and H (x,u) .Similarly, the map X → X v is an isomorphism between M x and V (x,u) .Each tangent vector Z ∈ (T M ) (x,u) can be written Z = X h + Y v , where X, Y ∈ M x are uniquely determined vectors.Horizontal and vertical lifts of vector fields on M can be defined in an obvious way and define vector fields on T M .
The Sasaki metric g s has been the most investigated among all possible Riemannnian metrics on T M .However, in many different contexts such metrics showed a very "rigid" behaviour.Moreover, g s represents only one possible choice inside a wide family of Riemannian metrics on T M , known as Riemannian g-natural metrics, which depend on several independent smooth functions from R + to R. As their name suggests, those metrics arise from a very "natural" construction starting from a Riemannian metric g over M .g-natural metrics arise from the description of all possible first order natural operators D : S 2 + T * (S 2 T * )T , transforming Riemannian metrics on manifolds into metrics on their tangent bundles, where S 2 + T * and S 2 T * denote the bundle functors of all Riemannian metrics and all symmetric (0, 2)-tensors over manifolds.For more details about the concept of naturality and related notions, we refer to [10].
We shall call g-natural metric a metric G on T M , coming from g by a first-order natural operator S 2 + T * (S 2 T * )T [2].Given an arbitrary g-natural metric G on the tangent bundle T M of a Riemannian manifold (M, g), there are six smooth functions α i , where r 2 = g x (u, u).For n = 1, the same holds with Convention a) In the sequel, all Riemannian g-natural metrics G on T M will be defined by the functions α i , b) Unless otherwise stated, all real functions α i , β i , φ i , α and φ and their derivatives are evaluated at r 2 := g x (u, u).c) We shall denote respectively by R and Q the curvature tensor and the Ricci operator of a Riemannian manifold (M, g).The tensor R is taken with the sign convention All examples presented in this paper will concern Riemannian g-natural metrics satisfying ) Note that, because of (2.1), α 2 = β 2 = 0 is equivalent to requiring that the horizontal and vertical distributions are mutually orthogonal with respect to G. On the other hand, inequalities in (2.3) are nothing but (2.2), rewritten when α 2 = β 2 = 0.The Levi-Civita connection ∇ of an arbitrary g-natural metric G on T M was written down in [3], to which we refer for more details.Restricting ourselves to the case when α 2 = β 2 = 0, we have the following Proposition 2.1.Let (M, g) be a Riemannian manifold, ∇ its Levi-Civita connection and R its curvature tensor.Let G be a Riemannian g-natural metric on T M , satisfying α 2 = β 2 = 0.Then, the Levi-Civita connection ∇ of (T M, G) is characterized by for all vector fields X, Y on M and (x, u) ∈ T M , where B, C and F are defined, for all u, X, Y ∈ M x , x ∈ M , by: where where where

Harmonicity of V : (M, g) → (T M, G)
We briefly recall that, given f : (M, g) → (M , g ) a smooth map between Riemannian manifolds, with M compact, the energy of f is defined as the integral where e(f ) = 1 2 ||f * || 2 = 1 2 tr g f * g is the energy density of f .With respect to a local orthonormal basis of vector fields {e 1 , .., e n } on M , the energy density is e(f ) = 1 2 n i=1 g (f * e i , f * e i ).Critical points of the energy functional on C ∞ (M, M ) are called harmonic maps.They have been characterized in [8] as maps having vanishing tension field τ (f ) = tr∇df .When M is not compact, a map f : (M, g) → (M , g ) is said to be harmonic if τ (f ) = 0.For more on harmonic maps, cf.[7], [15].Now, let (M, g) be a compact Riemannian manifold of dimension n and (T M, G) its tangent bundle, equipped with a g-natural Riemannian metric.Each vector field V ∈ X (M ) defines a smooth map V : (M, g) → (T M, G), p → V p ∈ M p .By definition, the energy E(V ) of V is the energy associated to the corresponding map V : (M, g) → (T M, G).Therefore, E(V ) = M e(V )dv g , where the density function e(V ) is given by {e 1 , .., e n } being any local orthonormal basis of vector fields defined in a neighborhood of p. Harmonicity of the map V : (M, g) → (T M, G) is studied in [1].Using formulas of Proposition 2.1, we have where r = ||V ||.
Next, considering the horizontal and vertical components of the tension field τ (V ) = tr(∇dV ) associated to the map V : (M, g) → (T M, G), harmonic maps V : (M, g) → (T M, G) where characterized in [1] for all Riemannian g-natural metrics on T M .Restricting ourselves to the case of Riemannian g-natural metrics satisfying (2.3), from the main result of [1] we have at once the following Theorem 3.1.Let (M, g) be a compact Riemannian manifold and G a Riemannian g-natural metric on T M , determined by functions α i , β i satisfying (2.3).A vector field V ∈ X(M ) defines a harmonic map V : (M, g) → (T M, G) if and only if As usual, we can assume condition τ (V ) = 0 as a definition of harmonic maps when M is not compact, and Theorem 3.1 extends at once to the non-compact case.
Remark 3.3.In the next Sections, we shall use Riemannian g-natural metrics on the tangent bundle T M , and the characterization obtained in this Section for their harmonic maps, in order to determine new examples of harmonic maps.There are some prices to pay when the Sasaki metric is replaced by Riemannian g-natural metrics: • With respect to these metrics, parallel vector fields are not always harmonic [1].

Harmonic maps
Let S n ⊂ R n+1 be the unit sphere with the induced Riemannian metric g.Xin [16] proved that for any compact manifold (N, h), all non-constant harmonic maps f : (S n , g) → (N, h) are unstable, provided that n ≥ 3.
Vector fields responsible for this unstability are the conformal gradient vector fields.Given any vector a ∈ R n+1 , a = 0, the conformal gradient vector field V a corresponding to a, is defined as V a = ∇λ a , where, for all x ∈ S n , λ a (x) =< x, a >.These vector fields satisfy [16]: Moreover, putting c 2 = ||a|| 2 , one has [5] Finally, (4.1) also easily implies Let now G be an arbitrary Riemannian g-natural metric on T S n .Using (4.1), (2.1) and (2.3), we obtain )}ω a ⊗ω a , (4.4) where ω a is the 1-form dual to V a .
Next, we recall that the volume of V a is defined as and the equality holds if and only if On the other hand, Sanini [14] proved that if f : (M, g) → (M , g ) is a conformal map and M is a compact two-dimensional manifold, then the energy of f remains unaltered for any deformation of the metric g on M .Therefore, we have the following Proposition 4.1.Let G be a Riemannian g-natural metric on T S 2 , satisfying Moreover, E(V a ) is stationary with respect to an arbitrary deformation of the standard metric g 0 on S 2 .
Harmonicity of conformal gradient vector fields was investigated in [5], equipping the tangent bundle with a metric of Cheeger-Gromoll type.However, in the examples found in [5], the metrics are not Riemannian, but have varying signature.Moreover, these metrics fail to provide examples of harmonic maps defined on S 2 .We will show in the sequel how to construct such examples using Riemannian g-natural metrics.
Consider a Riemannian g-natural metric G on T M of the special type described by (2.3).Moreover, we also assume β 1 = β 3 = 0. We then rewrite harmonicity conditions τ h (V ) = τ v (V ) = 0 for a conformal gradient vector field V a = ∇λ a .Using (4.1) and ( 4.3), we obtain that V a : S n → (T S n , G) is a harmonic map if and only if Calculating (α 1 + α 3 ) from the first equation in (4.5) and replacing into the second one, we see that if n = 2, then (4.5) reduces to its first equation, namely, which can be integrated finding solutions compatible with the Riemannian conditions (2.2).Thus, if G is a Riemannian g-natural metric on T S 2 determined by then V a : S 2 → (T S 2 , G) is a harmonic map.For example, we can take explicitly for any real constants µ > 0 and µ 1 ≥ 0. Note that if µ 1 = 0, then the Riemannian g-natural metric G determined by (4.8) is conformal to the Sasaki metric g s on T S 2 ; moreover, if G is a Riemannian g-natural metric on T S 2 determined by (4.7), then (by (4.4) ) V * a G is conformal to g. Next, we recall the following decreasing property for harmonic immersions of a surface, proved by Sampson ([13], Theorem 7, p.217): if f : (M 2 , g) → ( M , g) is a harmonic immersion and f * g is conformal to g, then the sectional curvatures of (M 2 , f * g) and ( M , g) satisfy for any x ∈ M 2 .This result ensures that (T S 2 , G) admits some positive sectional curvatures for any Riemannian g-natural metric G on T S 2 satisfying (4.7).In fact, the Gauss-Bonnet Theorem then gives where χ(S 2 ) denotes the Euler number of S 2 .Therefore, we proved the following Theorem 4.2.Let G be a Riemannian g-natural metric satisfying (4.7).
Then, all conformal gradient vector fields V a = ∇λ a on S 2 define harmonic maps V a : S 2 → (T S 2 , G).Moreover, (T S 2 , G) admits some positive sectional curvatures.
Remark 4.3.Standard calculations show that system (4.5), when n ≥ 3, admit solutions which depend on the function λ 2 a = c 2 − r 2 defining the conformal gradient vector field V a .Explicitly, we find + κ if n > 3, for arbitrary constants µ > 0 and κ.However, in general these solutions are not compatible with the Riemannian conditions (2.2).Indeed, an easy analysis of the above expressions of α 3 leads to the following conclusions: • The metric G is Riemannian if and only if n > 3, κ > 0 and • The metric G is pseudo-Riemannian of signature (n, n) if and only if either n = 3 and c 2 − e − κ µ ≤ 0, or n > 3 and κ ≤ 0.
• In the remaining cases, the metric G is of varying signature.
The following result has been proved in [4]: let f : (M 2 , g) → (M , g ) be a conformal harmonic map, with M compact.If f is volume-stable, then it is also energy-stable.Theorem 4.2 above then implies the following Corollary 4.4.Let G be a Riemannian g-natural metric satisfying (4.7).If the harmonic map V a : S 2 → (T S 2 , G) is volume-stable, then it is also energy-stable. 5. Harmonic maps V : (M (k), g) → (T M, G) when k ≤ 0 Let (M (k), g) be an n-dimensional Riemannian manifold of constant sectional curvature k ≤ 0, and V a vector field on M , of constant length √ ρ.Consider a Riemannian g-natural metric G on T M , determined by functions α i (t), β i (t) satisfying Using the second equation of (5.1), one can easily rewrite the connection functions given in Proposition 2.1 and then the horizontal and vertical components of the tension field τ (V ).We also use the constancy of the sectional curvature and of the length of V .We conclude that τ h (V ) = 0 identically, and Since in this case ∆V = 1 ρ ||∇V || 2 V , replacing into (5.3)we find In particular, assuming α 1 + α 3 = µ and taking α 1 (t) = ce − 1 ρ t , for two positive constants µ, c, (5.4) is automatically satisfied.Then, we get Theorem 5.1.Let (M (k), g) be an n-dimensional Riemannian manifold of nonpositive constant sectional curvature k, and G a Riemannian gnatural metric on T M , determined by functions α i (t), β i (t) which satisfy where ρ, µ, c are positive real constants.Then, any vector field V of constant lengt ρ defines a harmonic map V : (M (k), g) → (T M, G).Remark 5.2.Conditions (5.5) ensure that (2.2) are satisfied and so, gnatural metrics G as in Theorem 5.1 are Riemannian.Note that when k > 0, then (5.5) again determines g-natural metrics G for which all vector fields of constant length √ ρ are harmonic maps.However, these G are not Riemannian, because (2.2) are not satisfied.Explicit examples of nonparallel harmonic maps of (T M, G ), when (M, g) has positive constant sectional curvature and G is a suitable Riemannian g-natural metric, have already been found in [1].

Harmonic maps
Let (M, g) be an n-dimensional locally flat Riemannian manifold and consider a (local) orthonormal frame of coordinate vector fields As it is well known, if the tangent bundle T M is equipped with the Sasaki metric g s , then parallel vector fields are harmonic maps (M, g) → (T M, g s ).Moreover, when M = Ω is an open subset of R n , a vector field V = V i ∂ ∂x i defines a harmonic map from Ω → (T Ω, g s ) if and only if the map is harmonic.To our knowledge, these are the only general results on the harmonicity of vector fields defined in a locally flat manifold.Consider now M = Ω an open subset of R n , equipped with the induced flat metric, and a Riemannian g-natural metric G on T M , satisfying (2.3) and the additional conditions where a is a constant.Then, by Theorem 3.1 and taking into account R = 0 and (6.1), we have that (3.2) holds identically, and the map V : (M, g) → (T M, G) is harmonic if and only if Since V = V i ∂ ∂x i , we then have r 2 = i V 2 i and we easily find Hence, (6.2) is equivalent to the following system of partial differential equations: ∂V j ∂x i 2 V k = 0, k = 1, . . ., n.
(6.3)Because of the regularity of α 1 , general existence results for the solutions of a system of partial differential equations, ensure that system (6.3)admits solutions.Note that whenever at least one component V k is not constant, V is not parallel.
As an extremely special case, assume the components of V are all constant except for V k , for a fixed index k, and that V k only depends on one of the local coordinates, say V k = V k (x i ).Then, system (6.3)reduces to (6.4) Remark 6.1.It would be worthwhile to find some explicit solutions of system (6.3), also taking into account the fact that there exist some g-natural