Geodesics and associated Jacobi operators

P -spaces of Iwasawa type 59

2. Geodesics and associated Jacobi operators

Throughout this section and the following ones, s = n ⊕ a will denote a metric Lie algebra of Iwasawa type and algebraic rank one, where a = RH, |H | = 1, is chosen such that all eigenvalues of ad H | n are positive, and n = z ⊕ v is expressed as in the previous section. Let γ Y denote the geodesic in S satisfying γ Y 0 = e the identity of S and γ ′ Y 0 = Y. For any X ∈ s, the associated left invariant field along the geodesic γ Y will be denoted by X t = X γ Y t = dL γ Y t e X. Next, we compute the geodesic γ Y with Y ∈ n, an eigenvector of ad H | n . L EMMA 1. If Y ∈ n is an eigenvector of ad H | n with eigenvalue α, then γ Y t = exp n tanh αt α Y, − 1 α lncosh αt with associated tangent vector field γ ′ Y t = 1 cosh αt Y t − tanh αt H t. Proof. Let S be the simply connected Lie group associated to s , the Lie algebra spanned by {Y, H }. Note that S has global coordinates ϕx , r = exp n x Y, r and it is a totally geodesic subgroup of S with connection ∇ S = ∇| S satisfying ∇ Y Y = α H, ∇ Y H = −[H, Y ] = −αY, ∇ H = 0. Since the coordinate fields associated to ϕ are given by ∂ ∂ x ϕ x ,r = e − rα Y ϕx , r , ∂ ∂ r ϕ x ,r = H ϕx , r , the Christoffel symbols are easily computed by the formulas ∇ ∂ ∂ x ∂ ∂ x ϕ x ,r = α e − 2rα H ϕx , r , ∇ ∂ ∂ x ∂ ∂ r ϕ x ,r = −α ∂ ∂ x ϕ x ,r , ∇ ∂ ∂ r ∂ ∂ r ϕ x ,r = 0 = ∇ ∂ ∂ r ∂ ∂ x ϕ x ,r . Hence, we obtain the geodesic γ Y t = ϕx t, r t, where x t and r t are solutions of the differential equations x ′′ − 2αx ′ r ′ = 0, r ′′ + α e − 2rα x ′ 2 = 0. Using that γ ′ Y t = x ′ t ∂ ∂ x γ Y t + r ′ t ∂ ∂ r γ Y t , γ ′ Y t = 1, with x ′ t = e 2αrt , ∂ ∂ x γ Y t = e −α rt Y t and ∂ ∂ r γ Y t = H t, 60 M. J. Druetta we have the equivalent equations x ′ t = e 2αrt , r ′′ t + α1 − r ′ t 2 = whose solutions satisfy x t = Z t e 2αru du and r ′ t = − tanh αt. Therefore, we get r t = − 1 α lncosh αt, x t = 1 α tanh αt, and the expression of γ Y and γ ′ Y follows as claimed. P ROPOSITION 1. If Z ∈ z and X ∈ v are eigenvectors of ad H with associated eigenvalues λ and µ, respectively, then for any Y ∈ v, we have i R γ ′ Z t Y t = 1 cosh 2 λ t dL γ Z t e · R Z Y − sinh 2 λ t ad 2 H Y − sinh λt j Z 1 2 λ Id − ad H Y i i R γ ′ X t Z t = 1 cosh 2 µ t dL γ X t e · R X Z − λ 2 sinh 2 µ t Z + λ − 1 2 µ sinh µt j Z X i i i R γ ′ X t Y t = 1 cosh 2 µ t dL γ X t e · R X Y − sinh 2 µ t ad 2 H Y − sinh µt 1 2 µ Id − ad H [X, Y ] in the case of 2-step nilpotent n, with Y ⊥X in v. Proof. i Let Z ∈ z and γ Z t be the associated geodesic. Since γ ′ Z t = 1 cosh λt Z t − tanh λt H t, we have that RY t, γ ′ Z tγ ′ Z t = 1 cosh 2 λ t dL γ Z t e · R Z Y + sinh 2 λ t R H Y − sinh λt RY, Z H + RY, H Z . P -spaces of Iwasawa type 61 Using the Bianchi identity and the connection formulas we compute RY, Z H + RY, H Z = 2 RY, H Z − RZ , H Y = 2∇ [H,Y ] Z − ∇ [H,Z ] Y = − j Z ad H Y + 1 2 λ j Z Y, that substituted in the above expression, gives i as stated since R H = − ad 2 H . ii-iii Assume that X ∈ v is an eigenvector of ad H with eigenvalue µ, and let Y ⊥ X in v. Using the expression of γ ′ X t, in the same way as i we get RZ t, γ ′ X tγ ′ X t = 1 cosh 2 µ t dL γ X t e · R X Z + sinh 2 µ t R H Z − sinh µt RZ , X H + RZ , H X . Hence, the expression of R γ ′ X t Z t follows as claimed since RZ , X H + RZ , H X = 2∇ [H,Z ] X − ∇ [H,X ] Z = 2λ∇ Z X − µ∇ X Z = −λ − 1 2 µ j Z X. Finally, we have RY t, γ ′ X tγ ′ X t = 1 cosh 2 µ t dL γ X t e · R X Y + sinh 2 µ t R H Y − sinh µt RY, X H + RY, H X . In the same way as above, in the case of 2-step nilpotent n, we compute RY, X H + RY, H X = 2∇ [H,Y ] X − ∇ [H,X ] Y = − [H, [X, Y ]] + 1 2 µ [X, Y ] hX, Y i = 0, which completes the proof of the proposition.

3. Eigenvectors and eigenvalues along Jacobi operators