Fix a compact Riemannian manifold $ M$ , a base point $ p\in M$ , a basis $ B$ for the tangent space $ T_p$ , and a destination $ q$ .
The metric on $ M$ determines its Levi-Civita connection.
My goal consists of finding a short path from $ p$ to $ q$ of a special form: such a path must leave $ p$ traveling along a geodesic lying in a direction contained in $ B$ and, carrying $ B$ along according to the connection, arrive at new point $ p_1$ ; if $ p_1\not=q$ , then at $ p_1$ the path must turn in a new direction belonging the the transported basis. The steps then repeat until finally we arrive at $ q$ after a journey possessing a finite number of legs. Call such a path urban (because we must stay on the “city streets” determined by $ B$ ) unless the literature already has a name for this.
Clearly, with a bound on the number of legs, some urban path attains the infinimum length of all urban paths with no more than that many legs.
My questions: without a bound on the number of legs, must some urban path still attain the infinimum? If yes, can we get a bound, based solely on the dimension of $ M$ , for the number of legs in this “urban geodesic”?
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