On sub-Riemannian geodesic curvature in dimension three

Let Φ H → t denote the flow of H → . The map

is a diffeomorphism whose inverse is F - 1 : F ⁢ ( S ) → S , F - 1 ⁢ ( ξ , δ ¯ ) = Φ H → δ ¯ ⁢ ( ξ ) .

As a consequence of definitions (4.3) and (4.4), for every orthonormal frame of the distribution { X 1 , X 2 } and i , j , k = 0 , 1 , 2 , we have [ X ¯ i , X ¯ j ] = [ X i , X j ] ¯ , which in turn implies

A vector field 𝒥 defined along an integral line γ ¯ : I → T * ⁢ M of the Hamiltonian field is said to be a Jacobi field if the Lie derivative L H → ⁢ J = 0 along γ ¯ .

Let us consider { X 1 , X 2 } an orthonormal frame of the distribution. There exist two smooth vector fields J ⟂ : S → T ⁢ S and J 0 : S → T ⁢ S ,

that satisfy for i ∈ { ⟂ , 0 } the following conditions:

1. J i is a Jacobi field, i.e., [ J i , H → ] = 0 ;

2. for every ξ ∈ S p * ⁢ M , we have π * ⁢ ( J i ⁢ ( ξ ) ) = 0 ( J i is vertical at zero) and

   σ ⟂ ∘ F - 1 ⁢ ( ξ , δ ¯ ) ∼ δ ¯ 2 2   and   σ 0 ∘ F - 1 ⁢ ( ξ , δ ¯ ) ∼ - δ ¯ 3 6 .

Proof

By combining the expression of H → given by (4.5) and that of J i , we can reformulate the condition [ J i , H → ] = 0 by decomposing it on the frame

The corresponding system of differential equation is given by

In order to define the vector fields J ⟂ and J 0 , it is then sufficient to define their values on S p * ⁢ M , the values on the whole space F ⁢ ( S ) then following from the differential equation (A.3). We define, on S p * ⁢ M ,

We now use (A.3) to establish the asymptotics of σ 0 and σ ⟂ by computing the derivatives and evaluating at zero. We find out that, for any 𝜉 in S p * ⁢ M ,

Furthermore, using again ξ ∈ H - 1 ⁢ ( 1 / 2 ) ,

Now, thanks to the first identity in (A.1),

which proves the asymptotics in (ii). The fact that the functions σ i are smooth and independent of the choice of ( X 1 , X 2 ) is due to the identity σ i = ω ∘ d ⁢ π ⁢ ( J i ) . ∎

We have S ∖ S p * ⁢ M ⊂ h Γ - 1 ⁢ ( 1 ) ∩ h J ⁢ Γ - 1 ⁢ ( 0 ) .

Proof

Let us consider any covector in S ∖ S p * ⁢ M . It can be written as d ⁢ δ q for a certain 𝑞 in Σ p by definition of 𝔖. Now, if we choose ( Γ , J ⁢ Γ ) as a frame of the distribution, by using (4.1), we can write, thanks to Lemma 22,

The quantities δ ¯ ⁢ c Γ , J ⁢ Γ J ⁢ Γ and δ ¯ 2 ⁢ c Γ , X 0 J ⁢ Γ (a priori defined on S ∖ S p * ⁢ M ) can be smoothly extended to 𝔖 and are respectively equal to −4 and −6 over S p * ⁢ M .

Proof

Let us write the vector fields J 0 and J ⟂ introduced in Lemma 40 over S ∖ T p * ⁢ M with respect to the orthonormal frame Γ , J ⁢ Γ ,

Since S ∖ T p * ⁢ M is contained in h Γ - 1 ⁢ ( 1 ) ∩ h J ⁢ Γ - 1 ⁢ ( 0 ) (by Lemma 41), we have j 1 i = j 2 i = 0 , and the first and the third equation of (A.3) can be combined as 0 = H → 2 ⁢ σ i + c ¯ Γ , J ⁢ Γ J ⁢ Γ ⁢ H → ⁢ σ i + c ¯ X 0 , Γ J ⁢ Γ ⁢ σ i . Since this last equation is satisfied by σ 0 and σ ⟂ , we find out that

The matrix of this system as well as its left-hand side are smooth over 𝔖 when δ ¯ goes to zero by applying Lemma 26 to the asymptotics given in Lemma 40 (we use the first identity in (A.1)), and

Inverting (A.4), we obtain that the functions δ ¯ ⁢ c ¯ Γ , J ⁢ Γ J ⁢ Γ and δ ¯ 2 ⁢ c ¯ X 0 , Γ J ⁢ Γ that were a priori defined on S ∖ S p * ⁢ M can in fact be smoothly extended to the domain 𝔖. Taking then the limit as δ ¯ goes to zero, we find the values of δ ¯ ⁢ c ¯ Γ , J ⁢ Γ J ⁢ Γ and δ ¯ 2 ⁢ c ¯ X 0 , Γ J ⁢ Γ on the set δ ¯ - 1 ⁢ ( 0 ) = S p * ⁢ M . ∎

The function δ ¯ 2 ⁢ H → ⁢ c ¯ Γ , J ⁢ Γ J ⁢ Γ , a priori defined on S ∖ S p * ⁢ M , can be extended to a smooth function on 𝔖, and its evaluation is equal to 4 on S p * ⁢ M .

Proof

We know from Proposition 42 that δ ¯ ⁢ c ¯ Γ , J ⁢ Γ J ⁢ Γ can be extended to a smooth function on 𝔖 that is equal to −4 on δ ¯ - 1 ⁢ ( 0 ) . Since H → is also smooth, we can write

So, recalling that H → ⁢ δ ¯ , we have that

has a smooth extension on 𝔖 that is equal to 4 on δ ¯ - 1 ⁢ ( 0 ) = S p * ⁢ M . ∎

The function δ ¯ 2 ⁢ J ⁢ Γ ¯ ⁢ c ¯ Γ , J ⁢ Γ J ⁢ Γ that is a priori defined on S ∖ S p * ⁢ M can be extended to a smooth function on the domain 𝔖.

Proof

Let us consider the fields J 0 and J ⟂ over 𝔖 that we introduced in Lemma 40. We start by proving the following claim: for every 𝜉 in 𝔖, the vector V ⁢ ( ξ ) defined by (A.5) is colinear to J ⁢ Γ .

First notice that V ⁢ ( ξ ) belongs to the distribution since its component along the Reeb vector field X 0 is zero thanks to (A.2). Let us then prove that it is orthogonal to the gradient of 𝛿. Indeed, for i ∈ { 0 , ⟂ } , by definition of Jacobi field, one has 0 = [ H → , J i ] ⁢ δ ¯ = H → ⁢ J i ⁢ δ ¯ - J i ⁢ ( H → ⁢ δ ¯ ) = H → ⁢ J i ⁢ δ ¯ , where we used that H → ⁢ δ ¯ = 1 . So J i ⁢ δ ¯ is constant on the integral lines of H → . But, on S p * ⁢ M , the function J i ⁢ δ ¯ is equal to zero. Therefore, J i ⁢ δ ¯ = 0 on 𝔖. Thus d ⁢ π ⁢ ( J i ⁢ ( ξ ) ) belongs to the kernel of d ⁢ δ . The vector V ⁢ ( ξ ) is a linear combination of vectors in the kernel of d ⁢ δ ; hence it is also in the kernel of d ⁢ δ . By Lemma 22, this means that V ⁢ ( ξ ) is colinear to J ⁢ Γ , and the claim is proved.

Let us define b : S → R such that, for any 𝜉 in 𝔖,

in such a way that, for all 𝜉 in S ∖ T p * ⁢ M ,

Since c ¯ Γ , J ⁢ Γ J ⁢ Γ is constant on the fiber of T * ⁢ M , we can replace J ⁢ Γ ¯ in the expression with a vector field that projects over J ⁢ Γ . Then we have

To prove that the right-hand side of (A.7) is smooth, we write the fields J i over S ∖ T p * ⁢ M as

Combining with (A.5) and (A.6), we obtain that, for any 𝜉 in S ∖ T p * ⁢ M , b ⁢ ( ξ ) = σ ⟂ ⁢ ( ξ ) ⁢ β 0 ⁢ ( ξ ) - σ 0 ⁢ ( ξ ) ⁢ β ⟂ ⁢ ( ξ ) . Now, thanks to Lemma 41, h Γ = 1 and h J ⁢ Γ = 0 on S ∖ T p * ⁢ M , so the first equation of (A.3) becomes H → ⁢ σ i = - β i . As a consequence, for every 𝜉 in S ∖ T p * ⁢ M , b ⁢ ( ξ ) = σ 0 ⁢ ( ξ ) ⁢ H → ⁢ σ ⟂ ⁢ ( ξ ) - σ ⟂ ⁢ ( ξ ) ⁢ H → ⁢ σ 0 ⁢ ( ξ ) . Hence

where we used that J i ⁢ δ ¯ = 0 for i ∈ { 0 , ⟂ } . By applying Proposition 42, δ ¯ ⁢ c ¯ Γ , J ⁢ Γ J ⁢ Γ can be extended to a smooth function defined on 𝔖, and its value on S p * ⁢ M is constant. Therefore, the functions J i ⁢ ( δ ¯ ⁢ c ¯ Γ , J ⁢ Γ J ⁢ Γ ) can be extended to smooth functions on 𝔖 that vanish at every point of S p * ⁢ M . We combine this with the smoothness and the asymptotics of the functions σ i that come from Lemma 40, and thanks to Lemma 26, the function δ ¯ 2 ⁢ J ⁢ Γ ¯ ⁢ c Γ , J ⁢ Γ J ⁢ Γ can be extended to a smooth function on 𝔖. ∎