Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  develop a variational theory for geodesics joining a point and a one dimensional submanifold of a sub-Riemannian manifold. Given a Riemannian manifold (M,g), a smooth distribution $\Delta\subset T{M}$ of codimension one in M, a point $p\in{M}$, and a smooth immersion $\gamma:\R\rightarrow{M}$ with closed image in M and which is everywhere transversal to $\Delta$,  look for curves in M that are stationary with respect to the Riemannian energy functional among all of the absolutely continuous curves horizontal with respect to $\Delta$ and that join p and $\gamma$. If (M,g) is complete, such extremizers exist, and they are curves of class C2 characterized as the solutions of an integro-differential equation or by a system of ordinary differential equations.  present some results concerning a sort of exponential map relative to the integro-differential equation and some applications. In particular,  obtain that if p and $\gamma$ are sufficiently close in M, then there exists a unique length minimizer.  obtain existence and multiplicity results by means of the Ljusternik--Schnirelman theory.
['Roberto Giambò', 'Fabio Giannoni', 'Paolo Piccione']
I am writing a research paper on a idea Existence, Multiplicity, and Regularity for sub-Riemannian Geodesics by Variational Methods
[{"from": "human", "value": "I am writing a research paper on a idea Existence, Multiplicity, and Regularity for sub-Riemannian Geodesics by Variational Methods"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  develop a variational theory for geodesics joining a point and a one dimensional submanifold of a sub-Riemannian manifold. Given a Riemannian manifold (M,g), a smooth distribution $\\Delta\\subset T{M}$ of codimension one in M, a point $p\\in{M}$, and a smooth immersion $\\gamma:\\R\\rightarrow{M}$ with closed image in M and which is everywhere transversal to $\\Delta$,  look for curves in M that are stationary with respect to the Riemannian energy functional among all of the absolutely continuous curves horizontal with respect to $\\Delta$ and that join p and $\\gamma$. If (M,g) is complete, such extremizers exist, and they are curves of class C2 characterized as the solutions of an integro-differential equation or by a system of ordinary differential equations.  present some results concerning a sort of exponential map relative to the integro-differential equation and some applications. In particular,  obtain that if p and $\\gamma$ are sufficiently close in M, then there exists a unique length minimizer.  obtain existence and multiplicity results by means of the Ljusternik--Schnirelman theory."}]