8th Seminar on Geometry and Topology Amirkabir University of Technology, December 15-17, 2015
On the harmonic vector fields Amir Baghban∗ and Esmail Abedi
Abstract An isotropic almost complex structure Jδ,σ defines a Riemannian metric gδ,σ on the tangent bundle which is the generalized type of the Sasaki metric. In this paper, the harmonic maps defined by vector fields are studied where the tangent bundle will be equipped with the metric gδ,σ . AMS subject Classification 2010: 53C43, 53C15 Keywords: Energy functional, Isotropic almost complex structure,Tangent bundle, Tension field.
1
Introduction
Let (M, g) be a Riemannian manifold and T M be its tangent bundle. The most studied metrics on tangent bundles are the Sasaki metric gs and g-natural metrics G where g-natural metrics are the natural generalization of the Sasaki metric([4], [7], [11] and [13]). Moreover, there is an unknown class of metrics on tangent bundle induced by isotropic almost complex structures Jδ,σ defined by Aguilar [1] which are the generalization of the Sasaki metric. Investigating on harmonic vector fields is a topic which was studied when the tangent bundle is equipped with the Sasaki metric and g-natural metrics([2], [3], [5], [6], [8], [9],[10], [12], [14] and [15]). This problem goes back to find the critical points of the Dirichlet energy functional E : C ∞ ((M, g), (T M, h)) −→ R+ where h is an arbitrary Riemannian metric on T M and C ∞ ((M, g), (T M, h)) is the set of all smooth functions from (M, g) to (T M, h). When h is the Sasaki metric, parallel vector fields are the only ones which define a harmonic map from (M, g) to (T M, h). Moreover, when the Dirichlet energy functional is restrected to the set of all vector fields on M , parallel vector fields are also the only ones. But if we think of restrecting the energy functional on the set of all unit vector fields, then every unit vector field X is a critical point of such functional if and only if ∆g X = ||∇X||2 X, where ∆g is the so-called rough Laplacian with ∗
Speaker
1
2 respect to the metric g. In this paper, we equip the tangent bundle of M with the Riemannian metric gδ,σ and investigate the harmonic vector fields. The rest of this paper is organized as follows: Section 2 is devoted to the some basic informations on the tangent bundle and isotropic almost complex structures. In Section 3, we study the Riemannian metrics gδ,σ on T M and in particular, we calculate the Levi-Civita connection of these metrics. Section 4 elaborates on the tension field of a map X defined by an arbitrary vector field X : (M, g) −→ (T M, gδ,σ ) and moreover, the necassary conditions for the harmonicity of such maps are achived.
2
Preliminaries
Assume (M, g) is an n-dimensional Riemannian manifold and ∇ represents the Levi-Civita connection of g. Moreover, let π : T M −→ M be its tangent bundle and K : T T M −→ T M be the connection map with respect to ∇ where π is the natural projection. The tangent bundle of T M (T T M ) can be split to vertical and horizontal vector sub-bundles V and H, respectively, i.e., for every v ∈ T M , Tv T M = Vv ⊕ Hv . These sub-bundles have the following properties • π∗v |Hv : Hv −→ Tπ(v) M is an isomorphism. • K |Vv : Vv −→ Tπ(v) M is an isomorphism. Where π∗v is the differential map of π at v ∈ T M . Note that X v and X h , standing for the vertical and horizontal lifts of X , are the vector fields on TM defined by • X v (u) = (K |Vu )−1 Xπ(u) ∈ Vu . • X h (u) = (π∗v |Hu )−1 Xπ(u) ∈ Hu . where u ∈ T M . The Lie bracket of the horizontal and vertical vector fields at u ∈ T M are expressed as follows [X h , Y h ](u) = [X, Y ]h (u) − (R(X, Y )u)v (u),
(2.1)
[X h , Y v ](u) = (∇X Y )v (u),
(2.2)
v
v
[X , Y ](u) = 0(u).
(2.3)
Where 0 is the zero vector field on T M . Moreover, if we consider the vector field X : M −→ T M as a map between manifolds, its derivative X∗ at a point p in M is given by X∗p (V ) = V h (X(p)) + (∇V X)v (X(p))
∀V ∈ Γ(T M )
(2.4)
3
2.1
pullback bundle
Let f : (M m , g) −→ (N n , h) be a C ∞ function between Riemanian manifolds and ∇M and ∇N be the Levi-Civita connections of g and h, respectively. If T N be the tangent bundle of N , then we can define vector bundle f −1 T N on M by (f −1 T N )p = Tf (p) N
∀p ∈ M.
There is a natural connection on f −1 T N defined by (f −1 ∇N )
∂ ∂xi
Yβ =
∂f α N γ [(Γ )αβ of ]Yγ , ∂xi
for all i = 1, ..., m and α, β, γ = 1, ..., n. Where (x1 , ..., xm ) and (y1 , ..., yn ) are the local charts on M and N, respectively, {Y1 = ∂y∂ 1 of, ..., Yn = ∂y∂n of } is a local basis for the sections of f −1 T N , {(ΓN )γαβ } are the Christoffel symbols of the Levi-Civita connection of h, f = (f1 , ..., fn ) stands for the locally coordinate of f on the domain of (x1 , ..., xm ).
2.2
Isotropic almost complex structures
Isotropic almost complex structures are a generalized type of the natural almost complex structure J1,0 : T T M −→ T T M given by J1,0 (X h ) = X v ,
J1,0 (X v ) = −X h
∀X ∈ Γ(T M ).
(2.5)
The isotropic almost complex structures determine an almost kahler metric which kahler 2-form is the pullback of the canonical symplectic form on T ∗ M to T M via b : T M −→ T ∗ M . Moreover, it is proved [1] that there is an isotropic complex structure on M if and only if M is of constant sectional curvature. Definition 2.1. [1] An almost complex structure J on T M is said to be isotropic with respect to the Riemannian metric on M , if there are smooth functions α, δ, σ : T M −→ R such that αδ − σ 2 = 1 and JX h = αX v + σX h ,
JX v = −σX v − δX h
∀X ∈ Γ(T M ).
(2.6)
Hereafter, we will represent the isotropic almost complex structure associated to the maps α, δ and σ with Jδ,σ .
3
A class of Riemannian metrics on T M
In this section, the Riemannian metric associated to the almost complex structure Jδ,σ will be introduced. Let Θ ∈ Ω1 (T M ) be a 1-form on T M defined by Θv (A) = gπ(v) (π∗ (A), v),
A ∈ Tv T M,
v ∈ T M.
(3.1)
4 Definition 3.1. [8] Let Jδ,σ be an isotropic almost complex structure. (0,2)-tensor gδ,σ (A, B) = dΘ(Jδ,σ A, B) defines a Riemannian metric on T M provided that α > 0 where A, B ∈ Γ(T T M ). Let X, Y be local sections of T M . A simple calculation shows that gδ,σ (X h , Y h ) =αg(X, Y )oπ,
(3.2)
gδ,σ (X h , Y v ) = − σg(X, Y )oπ,
(3.3)
v
v
gδ,σ (X , Y ) =δg(X, Y )oπ.
(3.4)
To calculate the Levi-Civita connection of gδ,σ , we need the following lemma. Lemma 3.2. [4] Let X, Y and Z be any vector fields on M . Then X h (g(Y, Z)oπ) = (Xg(Y, Z))oπ, v
X (g(Y, Z)oπ) = 0.
(3.5) (3.6)
Theorem 3.3. Let gδ,σ be a Riemannian metric on T M as above. Then the Levi-Civita con¯ of gδ,σ at (p, u) ∈ T M is given by nection ∇ ¯ X h Y h = (∇X Y )h − σ (R(u, X)Y )h + 1 X h (α)Y h + 1 Y h (α)X h ∇ α 2α 2α σ 1 1 h v v v − (∇X Y ) − (R(X, Y )u) − X (σ)Y δ 2 2δ 1 h 1 ¯ − Y (σ)X v − g(X, Y )∇α, (3.7) 2δ 2 ¯ X h Y v = − σ (∇X Y )h + δ (R(u, Y )X)h − 1 X h (σ)Y h ∇ α 2α 2α 1 v 1 1 + Y (α)X h + (∇X Y )v + X h (δ)Y v − Y v (σ)X v 2α 2δ 2δ 1 ¯ + g(X, Y )∇σ, (3.8) 2 ¯ X v Y h = δ (R(u, X)Y )h + 1 X v (α)Y h − 1 Y h (σ)X h ∇ 2α 2α 2α 1 v 1 h 1 v v ¯ − X (σ)Y + Y (δ)X + g(X, Y )∇σ, (3.9) 2δ 2δ 2 ¯ X v Y v = − 1 X v (σ)Y h − 1 Y v (σ)X h + 1 X v (δ)Y v ∇ 2α 2α 2δ 1 v 1 v ¯ + Y (δ)X − g(X, Y )∇δ. (3.10) 2δ 2 Proof. We just prove (3.7), the remaining ones are similar. Using Koszul formula, we have ¯ X h Y h , Z h ) = X h gδ,σ (Y h , Z h ) + Y h gδ,σ (X h , Z h ) − Z h gδ,σ (X h , Y h ) 2gδ,σ (∇ + gδ,σ ([X h , Y h ], Z h ) + gδ,σ ([Z h , X h ], Y h ) − gδ,σ ([Y h , Z h ], X h ).
5 Using relations (2.1), (3.2) and (3.5) gives us ¯ X h Y h , Z h ) = X h (α)g(Y, Z) + αXg(Y, Z) + Y h (α)g(X, Z) 2gδ,σ (∇ + αY g(X, Z) − Z h (α)g(X, Y ) − αZg(X, Y ) + αg([X, Y ], Z) + σg(R(X, Y )u, Z) + αg([Z, X]Y ) + σg(R(Z, X)u, Y ) − αg([Y, Z], X) − σg(R(Y, Z)u, X). Using the properties of the Levi-Civita connection of g, one can get ¯ X h Y h , Z h ) = g(X h (α)Y, Z) + g(Y h (α)X, Z) − Z h (α)g(X, Y ) 2gδ,σ (∇ + 2αg(∇X Y, Z) + σg(R(X, Y )u, Z) + σg(R(Z, X)u, Y ) − σg(R(Y, Z)u, X). Taking into account (3.2) and the Bianchis first identity, we have ¯ ¯ X h Y h , Z h ) = gδ,σ ( 1 X h (α)Y h + 1 Y h (α)X h − g(X, Y )∇α 2gδ,σ (∇ α α 2σ + 2(∇X Y )h − (R(u, X)Y )h , Z h ), α ¯ X h Y h is so the horizontal component of ∇ 1 h 1 1 h ¯ + (∇X Y )h X (α)Y h + Y (α)X h − g(X, Y )h(∇α) 2α 2α 2 σ − (R(u, X)Y )h , α
¯ Xh Y h) = h(∇
¯ = h(∇α) ¯ + v(∇α) ¯ where ∇α is the splitting of the gradient vector field of α with respect to gδ,σ ¯ Xh Y h to horizontal and vertical components, respectively. Similarly the vertical component of ∇ is 1 h 1 1 ¯ X (σ)Y v − Y h (σ)X v − g(X, Y )v(∇α) 2δ 2δ 2 σ 1 − (∇X Y )v − (R(X, Y )u)v . δ 2
¯ Xh Y h) = − v(∇
¯ X h Y h ) = h(∇ ¯ X h Y h ) + v(∇ ¯ X h Y h ), the proof is completed. Using the equation (∇
4
Hormonicity of a map X : (M, g) −→ (T M, gδ,σ ) defined by the vector field X
In this section, the definition of the tension field associated to a map between Riemannian manifolds is retrieved from [8]. So, one can refer to [8] for more details.
6 Suppose (M, g) and (M ′ , g ′ ) are Riemannian manifolds, with M compact. The Dirichlet energy associated to the Riamannian manifolds (M, g) and (M ′ , g ′ ) is defined by E :C ∞ (M, M ′ ) −→ R+ ∫ 1 f 7−→ ||df ||2 dvol(g). 2 M
(4.1)
Where ||df || is the Hilbert-Schmitd norm of df , i.e., ||df ||2 = trg (f ∗ g ′ ) and dvol(g) is the Riemannian volume form on M with respect to g. Remark 4.1. The expression e(f ) = 12 ||df ||2 = 12 trg (f ∗ g ′ ) is the so-called energy density of f . The critical points of E are defined as harmonic maps. It is proved [16] that a map f : (M, g) −→ (M ′ , g ′ ) is a harmonic map if and only if the tension field associated to f vanishes identically. Therefore, one can investigate the harmonicity of a map defined by a vector field by calculating the tension field associated to this map. Suppose (M, g) is a compact Riemannian manifold and W ∈ Γ(T M ). Let {V1 , ..., Vn } be a local orthonormal basis for the vector fields, defined in a neighborhood of p ∈ M which ∇Vi = 0 at p. The Dirichlet energy of the map W : (M, g) −→ (T M, gδ,σ ) defined by W is calculated as follow ∫ ∫ 1 1 E(W ) = ||dW ||2 dvol(g) = trg W ∗ (gδ,σ )dvol(g) 2 M 2 M ∫ ∑ n 1 gδ,σ (W∗ (Vi ), W∗ (Vi ))dvol(g) = 2 M i=1 ∫ ∑ n 1 = gδ,σ (Vih + (∇Vi W )v , Vih + (∇Vi W )v )dvol(g) 2 M ∫ i=1 1 = (nα − 2σdiv(W ) + δ||∇W ||2 )dvol(g). (4.2) 2 M The tension field associated to the map X : (M, g) −→ (T M, gδ,σ ) is defined as follow τ (X) : M −→ X −1 (T T M ) where locally defined by τq (X) =
n ∑
¯ X (V ) X∗ (Vi ) − X∗ (∇V Vi )}(X(q)), {∇ i ∗ i
(4.3)
i=1
for every q in the domain of Vi , i = 1, ..., n. This definition is independent of the choice of {V1 , ..., Vn } so is a global definition on M .
7 Therefore, one can write τq (X) =
n ∑
¯ X (V ) X∗ (Vi ) − X∗ (∇V Vi )}(X(q)) {∇ i ∗ i
i=1
=
n ∑
¯ h {∇ V +(∇V i
i
h X)v Vi
+ (∇Vi X)v
i=1
− (∇Vi Vi )h − (∇∇Vi Vi X)v }(X(q)) =
n ∑
¯ hV h + ∇ ¯ h (∇V X)v + ∇ ¯ (∇ {∇ i i V V V i
i
i
h X)v Vi
i=1
¯ (∇ +∇ V
i
v X)v (∇Vi X)
− (∇Vi Vi )h − (∇∇Vi Vi X)v }(X(q)).
(4.4)
By using the equations (3.7),...,(3.10) and (∇Vi )(p) = 0 in (4.4), we have the following formula of the tension field associated to X 1 nα α {(1 − )X1 − ||∇X||2 Y1 + αdiv(X)Z1 + trg (∇. X)v (α). α 2 2 − σRic(X) − ∇αZ1 −σZ2 X − trg (∇. X)v (σ)∇. X
τp (X) =
− σtrg (∇. ∇. X) + δtrg R(X, ∇. X).}h (X(p)) 1 nδ δ + {− X2 − ||∇X||2 Y2 + δdiv(X)Z2 − αZ1 + σZ2 δ 2 2 − trg (∇. X)v (σ). + ∇αY1 −σY2 X + trg (∇. X)v (δ)∇. X + δ∆g X}v (X(p))
(4.5)
¯ ¯ ¯ ¯ Where X1 = π∗ ((∇α)oX), X2 = K((∇α)oX), Y1 = π∗ ((∇δ)oX), Y2 = K((∇δ)oX), Z1 = ¯ ¯ π∗ ((∇σ)oX) and Z2 = K((∇σ)oX) are vector fields on M ∑ defined by the compositions of the gradient vector fields of α, δ and σ with X and ∆g X = − ni=1 {∇Vi ∇Vi X − ∇∇Vi Vi X} is the so-called rough Laplacian of X. Remark 4.2. All of the traces stated in the equation (4.5) are the traces of some tensors, so are independent of the choice of the orthonormal frame. Therefore, for the map X : (M, g) −→ (T M, gδ,σ ) which is defined by a vector field X, one can get the following theorem Theorem 4.3. Suppose (M, g) is a compact Riemannian manifold and (T M, gδ,σ ) is its tangent bundle equipped with an arbitrary Riemannian metric gδ,σ as before. Then the map X : (M, g) −→ (T M, gδ,σ ) defined by the vector field X is a harmonic map if and only if nα α )X1 − ||∇X||2 Y1 + αdiv(X)Z1 + trg (∇. X)v (α). 2 2 − σRic(X) − ∇αZ1 −σZ2 X − trg (∇. X)v (σ)∇. X
(1 −
− σtrg (∇. ∇. X) + δtrg R(X, ∇. X). = 0
(4.6)
8 and nδ δ X2 − ||∇X||2 Y2 + δdiv(X)Z2 − αZ1 + σZ2 2 2 − trg (∇. X)v (σ). + ∇αY1 −σY2 X + trg (∇. X)v (δ)∇. X
−
+ δ∆g X = 0
(4.7)
Remark 4.4. Since the condition τ (X) = 0 has a tensorial character, as usual we can assume it as a definition of harmonic maps even when M is not compact. By supposing σ = 0 on T M and X1 = X2 = 0 in (4.6) and (4.7), we can deduce that (4.6) and (4.7) are reduced to the well-known result for the Sasaki metric, That is, the necessary and sufficient conditions for a vector field to be a harmonic map from (M, g) to (T M, gs ) i.e., the following equations tr[R(X, ∇. X).] = 0
and
∆g X = 0.
(4.8)
Note that in the case that M is a compact manifold, ∆g X = 0 implies that the equation tr[R(X, ∇. X).] = 0 holds. Moreover, the conditions X1 = X2 = 0 do not make that α be a constant function throughout T M . So, there are a class of Riemannian metrics gδ,0 on T M containing the Sasaki metric such that the sufficient and necessary conditions for the harmonicity of a map defined by a vector field X : (M, g) −→ (T M, gδ,0 ) is the equations (4.8). As a corollary of the theorem 4.3, When T M is equipped with the metric gδ,0 then a parallel vector field X defines a harmonic map X : (M, g) −→ (T M, gδ,0 ) when (1 − nα 2 )X1 = 0 and X2 = 0. Example 4.5. We suppose that (M, g) is the Euclidean space (Rn , ⟨., .⟩). Let (x1 , ..., xn ) be the standard coordinate system on Rn and ∂x∂ 1 ,..., ∂x∂n be the canonical vector fields on Rn associated to the x1 , ..., xn . We now define α : T M −→ R as follow α(v) = ⟨
∂ , v⟩2 + 1, ∂x2
(4.9)
for every v ∈ T Rn . It is obvious that α is a non vanishing positive function on T Rn which is not constant throughout T Rn . So, gδ,0 is a Riemannian metric on T M where δ is given by δ = α1 . If (x1 , ..., xn , y1 , ..., yn ) is the standard coordinate system on T M associated to (x1 , ..., xn ), then, one can get dv α = 2v2 dy2 ,
(4.10)
where v = (v1 , ..., vn ) is a vector in T M and dα was calculated at this point. Moreover, with ¯ at the point v = vi ∂ is defined as follow respect to the metric gδ,0 , ∇α ∂xi ¯ (∇α)(v) = 2v2 α(v)
∂ (v). ∂y2
(4.11)
9 ¯ ¯ Let X = ∂x∂ 1 . It is easy to see that X1 = π∗ ((∇α)oX) = 0 and X2 = K((∇α)oX) = 0. Since ∂ ∇X = 0 , the equations (4.6) and (4.7) hold. So X = ∂x1 is a harmonic map from (Rn , ⟨., .⟩) to (T Rn , gδ,0 ). Note that because of the remark 4.4, it is not necessary to M be a compact manifold in the above example.
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10 [13] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10 (1958), 338-354. [14] G. Wiegmink, Total bending of vector fields on Riemannian manifolds, Math. Ann. 303(1995), no.2, 325-344. [15] C. M. Wood, On the energy of a unit vector field, Geometriae Dedicata, 64 (1997),319-330. [16] H. Urakawa, Calculus of variations and harmonic maps, American Mathematical Society, 1993. Amir Baghban Department of Mathematics, Faculty of Science Azarbaijan Shahid Madani University Tabriz 53751 71379, Iran E-mail:
[email protected] Esmail Abedi Department of Mathematics, Faculty of Science Azarbaijan Shahid Madani University Tabriz 53751 71379, Iran E-mail:
[email protected]
