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29
The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups
, 2008
"... We present an invariant definition of the hypoelliptic Laplacian on subRiemannian structures with constant growth vector using the Popp’s volume form introduced by Montgomery. This definition generalizes the one of the LaplaceBeltrami operator in Riemannian geometry. In the case of leftinvariant ..."
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Cited by 38 (10 self)
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We present an invariant definition of the hypoelliptic Laplacian on subRiemannian structures with constant growth vector using the Popp’s volume form introduced by Montgomery. This definition generalizes the one of the LaplaceBeltrami operator in Riemannian geometry. In the case of leftinvariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares. We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.
SubRiemannian structures on 3D Lie groups
 J. Dynam. Control Systems
"... We give a complete classification of leftinvariant subRiemannian structures on three dimensional Lie groups in terms of the basic differential invariants. As a corollary we explicitly find a subRiemannian isometry between the nonisomorphic Lie groups SL(2) and A+(R) × S1, where A+(R) denotes the ..."
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Cited by 14 (4 self)
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We give a complete classification of leftinvariant subRiemannian structures on three dimensional Lie groups in terms of the basic differential invariants. As a corollary we explicitly find a subRiemannian isometry between the nonisomorphic Lie groups SL(2) and A+(R) × S1, where A+(R) denotes the group of orientation preserving affine maps on the real line. 1
Maxwell strata in subRiemannian problem on the group of motions of a plane
, 2008
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BISHOP AND LAPLACIAN COMPARISON THEOREMS ON THREE DIMENSIONAL CONTACT SUBRIEMANNIAN MANIFOLDS WITH SYMMETRY
"... Abstract. We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three dimensional contact subriemannian manifolds with symmetry. 1. ..."
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Cited by 8 (3 self)
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Abstract. We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three dimensional contact subriemannian manifolds with symmetry. 1.
SubRiemannian and subLorentzian geometry on SU(1, 1) and on its universal
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Small time heat kernel asymptotics at the subRiemannian cut locus
 J. Differential Geometry
"... For a subRiemannian manifold provided with a smooth volume, we relate the small time asymptotics of the heat kernel at a point y of the cut locus from x with roughly “how much ” y is conjugate to x. This is done under the hypothesis that all minimizers connecting x to y are strongly normal, i.e. al ..."
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Cited by 4 (1 self)
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For a subRiemannian manifold provided with a smooth volume, we relate the small time asymptotics of the heat kernel at a point y of the cut locus from x with roughly “how much ” y is conjugate to x. This is done under the hypothesis that all minimizers connecting x to y are strongly normal, i.e. all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre 4t log pt(x, y) → −d2(x, y) for t → 0, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the subRiemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the subRiemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane endowed with the Euclidean volume we get the expansion pt(x, y) ∼ t−5/4 exp(−d2(x, y)/4t) where y is reached from a Riemannian point x by a minimizing geodesic which is conjugate at y. 1
On 2step, corank 2 nilpotent subRiemannian metrics
 SIAM J. CONTROL OPTIM
, 2012
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An Invitation to CauchyRiemann and SubRiemannian Geometries
"... Acertain amount of complex analysis, in both one and several variables, combines with some differential geometry to form the basis of research in diverse parts of mathematics. We first describe that foundation and then invite the reader to follow specific geometric paths based upon it. We discuss su ..."
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Cited by 4 (0 self)
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Acertain amount of complex analysis, in both one and several variables, combines with some differential geometry to form the basis of research in diverse parts of mathematics. We first describe that foundation and then invite the reader to follow specific geometric paths based upon it. We discuss such topics as the failure of the Riemann mapping theorem in several variables, the geometry of real hypersurfaces in complex Euclidean space, the Heisenberg group, subRiemannian manifolds and their metric space structure, the Hopf fibration, subRiemannian geodesics on the threesphere, CR mappings invariant under a finite group, and finite type conditions. All of these topics fall within the branch of mathematics described by the title, but of course they form only a small part of it. We hope that the chosen topics are representative and appealing. The connections among them provide a fertile ground for future research. We modestly hope that this article inspires others to develop these connections further. To this end, our reference list includes a diverse collection of books and accessible articles. Given Riemann’s contributions to geometry, it is not surprising that his name occurs twice in the title. It is more surprising that the uses of his name here come from different parts of mathematics. We find it delightful that contemporary mathematics is forging connections that even Riemann could not have anticipated.
Riemannian metrics on 2dmanifolds related to the EulerPoinsot rigid body motion
 ESAIM Control
, 2014
"... Abstract. The EulerPoinsot rigid body motion is a standard mechanical system and it is a model for leftinvariant Riemannian metrics on SO(3). In this article using the SerretAndoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluati ..."
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Abstract. The EulerPoinsot rigid body motion is a standard mechanical system and it is a model for leftinvariant Riemannian metrics on SO(3). In this article using the SerretAndoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover, the metric can be restricted to a 2Dsurface, and the conjugate points of this metric are evaluated using recent works on surfaces of revolution. Another related 2Dmetric on S2 associated to the dynamics of spin particles with Ising coupling is analysed using both geometric techniques and numerical simulations. AMS Subject Classification. 49K15, 53C20, 70Q05, 81Q93.