An intrinsic measure for submanifolds in stratified groups

Magnani, Valentino; Vittone, Davide

doi:10.1515/crelle.2008.044

Cited by 35 publications

(109 citation statements)

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“…From the development of these tools, we naturally meet the notion of regular set modeled on a couple of groups. A discussion on this project can be found in the first version of this work [20].…”

Section: Introductionmentioning

confidence: 99%

Towards Differential Calculus in Stratified groups

Magnani¹

2013

J. Aust. Math. Soc.

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We study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.2010 Mathematics subject classification: primary 22E30; secondary 26B10, 26B12.

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Section: Introductionmentioning

confidence: 99%

Towards Differential Calculus in Stratified groups

Magnani¹

2013

J. Aust. Math. Soc.

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“…We limit ourselves to mentioning just a few recent works for a sketchy overview of this subject and further references, [1], [2], [4], [5], [8], [10], [11], [18], [19], [29], [33], [34], [35].…”

Section: Introductionmentioning

confidence: 99%

“…Observe that a non-horizontal submanifold may have horizontal points. These two classes of submanifolds were introduced recently in [33], whose results can be applied especially to horizontal submanifolds.…”

Section: Introductionmentioning

confidence: 99%

“…This formula provides the "natural" intrinsic measure of non-horizontal submanifolds, according to [33] and then provides an integral formula for the (Q−k)-dimensional spherical Hausdorff measure of these C 1 smooth k-codimensional submanifolds. As a byproduct, formula (3) also shows in particular that the Hausdorff dimension of smooth non-horizontal submanifolds is equal to Q−k.…”

Section: Introductionmentioning

confidence: 99%

“…In Definition 3.3, we recall this notion, which was introduced in [33] to study the intrinsic measure of higher codimensional submanifolds. In rough terms, the projection (v) Q−k represents the component of v having weight equal to Q−k with respect to intrinsic dilations of the group.…”

Section: Introductionmentioning

confidence: 99%

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Non-horizontal submanifolds and coarea formula

Magnani

2008

J Anal Math

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Let k be a positive integer and let m be the dimension of the horizontal subspace of a stratified group. Under the condition k ≤ m, we show that all submanifolds of codimension k are generically non-horizontal. For these submanifolds, we prove an area-type formula that allows us to compute their Q − k dimensional spherical Hausdorff measure. Finally, we observe that a.e. level set of a sufficiently regular vector-valued mapping on a stratified group is a non-horizontal submanifold. This allows us to establish a sub-Riemannian coarea formula for vector-valued Riemannian Lipschitz mappings on stratified groups.

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Submanifolds of Fixed Degree in Graded Manifolds for Perceptual Completion

Citti

Giovannardi

Ritoré

et al. 2021

Lecture Notes in Computer Science

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We extend to a Engel type structure a cortically inspired model of perceptual completion initially proposed in the Lie group of positions and orientations with a sub-Riemannian metric. According to this model, a given image is lifted in the group and completed by a minimal surface. The main obstacle in extending the model to a higher dimensional group, which can code also curvatures, is the lack of a good definition of codimension 2 minimal surface. We present here this notion, and describe an application to image completion.

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An intrinsic measure for submanifolds in stratified groups

Cited by 35 publications

References 35 publications

Towards Differential Calculus in Stratified groups

Towards Differential Calculus in Stratified groups

Non-horizontal submanifolds and coarea formula

Submanifolds of Fixed Degree in Graded Manifolds for Perceptual Completion

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