Loop spaces in subRiemannian manifolds
with Agrachev and Gentile.
Let be a subRiemannian manifold. How complicated is the space of admissible curves joning and with energy ?
By complicated, I mean topologically complicated, as measured by the sum of Betti numbers, .
Example 1 (Heisenberg group) Let be an orthonormal frame of the contact structure. Energy of a curve such that is .
1. Morse Theory
It implies that the Betti sum critical points with . In our case, critical points are geodesics joining and .
In Heisenberg group, geodesics from origin to come in one parameter families. For every integer , there is a circle of geodesics from origin to with energy . So critical points with form a disjoint union of circles. This set has Betti sum which is linear in . On the other hand, it turns out that is a sphere, its Betti sum equals 2.
We shall see, more generally, for step 2 Carnot groups, that the Betti sum of the set of geodesics increases like where is the co-rank, whereas the Betti sum of the set of all admissible curves grows like .
Theorem 1 Let be a 2-step Carnot group. For a generic final point, is a Hilbert manifold homeomorphic to the infinite dimensional sphere.
2.1. Structure of geodesics
Fact. Let be a 2-step Carnot group, with (should persist if . If the final point is not vertical and generic, there are only a finite number of geodesics from to .
Therefore, we shall concentrate on vertical final points.
A 2-step Carnot group is determined by a vectorspace of skew-symmetric matrices. Denote by
This a countable collection of algebraic sets. View .
Theorem 2 Geodesics appear in families . Each family is labelled by one of the points in sphere of radius where is orthogonal to the sphere…. The energy of the family is . The family is a product of at most copies of .
Corollary 3 .
2.2. Complexity of admissible curves
Theorem 4 . So Morse inequalities are not sharp.
Consider the end-point map in the Heisenberg case. Ending in the vertical direction is one equation (average zero). Morse theory (backwards) and infinite negative index implies one need only study curves with energy equal . Endpoint map in quadratic. For a quadratic map on the spehere, real algebraic geometry tells us that complexity grows polynomially of the announced degree.
Theorem 5 . Furthermore, can be computed.
In that case, dim. where are skew-symmetric matrices. has eigenvalues … and is given by some integral.
2.4. Role of self-similarity
Applying dilations, we see that . Thus as tends to and energy bound decreases quadratically in , our formula gives the derivative of complexity as a function of .
Note that when , has the same homotopy as , i.e. trivial. And there is only one geodesic, the constant.