Point Classification of Second Order ODEs: Tresse Classification Revisited and Beyond

Kruglikov, Boris

doi:10.1007/978-3-642-00873-3_10

Cited by 29 publications

(34 citation statements)

References 20 publications

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“…For a more detailed study of such transformations, from a general point of view, see e.g. [20,28]. Let us consider a general (2-dimensional) projective connection,…”

Section: Metrizable Projective Connections Degree Of Mobility and LImentioning

confidence: 99%

Normal forms of two-dimensional metrics admitting exactly one essential projective vector field

Manno

Vollmer

2020

Journal de Mathématiques Pures et Appliquées

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We give a complete list of mutually non-diffeomorphic normal forms for the two-dimensional metrics that admit one essential (i.e., non-homothetic) projective vector field. This revises some results in [25] and extends the results of [10,25], solving a problem posed by Sophus Lie in 1882 [22]. MSC 2010 classes: 53A20, 53A55, 53B10 Keywords: projective connections; projective symmetries; projectively equivalent metrics Problem 1 (Lie, 1882). Determine the metrics that describe surfaces whose geodesic curves admit an infinitesimal transformation 1 , i.e. metrics whose projective algebra has dim(p(g)) ≥ 1.Fubini referred to this problem as the "Lie problem", see [1,16]. An overview of the history of the problem can be found, for instance, in [1,2]. Important works in the field, with respect to the considerations in the present paper, are for instance [14,23,10,25,9].Problem 1 is also referred to as Lie's First Problem, in contrast to the narrower problem when the inequality dim(p(g)) > 1 holds strictly. The latter problem is then referred to as Lie's Second Problem. A solution to Lie's (Second) Problem was first claimed in [1,2], but the proof contained a gap. A correct solution is given in [10], where mutually non-diffeomorphic normal forms of 2-dimensional metrics g with dim(p(g)) ≥ 2 were found around generic points (i.e., the orbit of the projective algebra is of constant non-zero dimension in a neighborhood of these points). Metrics admitting exactly one, essential projective vector field have been treated in [25], around generic points, providing an explicit list of projective classes. However, this list is not a list of mutually non-diffeomorphic normal forms, i.e. non-isometric metrics; actually, it is not even sharp as a list of projective classes under projective transformations. Moreover, reference [25] contains some gaps and some supplementary results are incomplete, see Section 2.1 where these issues are discussed in detail, along with a brief outline of the state of art. The main outcomes of the present paper are discussed in Section 2.2: Theorems 2, 3 and 4 together with Proposition 5, constitute a classification in terms of normal forms up to isometries of metrics that admit exactly one, essential projective vector field. Note that Theorems 2 and 3 constitute corrections of two results found in [25], namely Theorems 2 and 3 of this reference. As a by-product of the proof of Theorem 4, we also obtain a classification of all projective classes that cover metrics with exactly one, projective vector field that is essential (as stated above, [25] provides only a description of such classes, not a classification, see Section 2.1 for more details).For the formulation of the main results, we need the following proposition.1 German original [22]: "Es wird verlangt, die Form des Bogenelementes einer jeden Fläche zu bestimmen, deren geodätische Curven eine infinitesimale Transformation gestatten." 2 In the current context, we use Proposition 1 to describe a pair of projectively equivalent metrics that serve as "...

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“…For a more detailed study of such transformations, from a general point of view, see e.g. [20,28]. Let us consider a general (2-dimensional) projective connection,…”

Section: Metrizable Projective Connections Degree Of Mobility and LImentioning

confidence: 99%

Normal forms of two-dimensional metrics admitting exactly one essential projective vector field

Manno

Vollmer

2020

Journal de Mathématiques Pures et Appliquées

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“…The problem of differential invariants of this action was initiated by S. Lie [27], and all relative invariants were found by A. Tresse [40]. The absolute differential invariants were derived and counted in [16]: h k = 0 for k ≤ 4, h 5 = 3 and h k = k 2 − 4 for k > 5. Therefore we obtain…”

Section: A Panorama Of Examplesmentioning

confidence: 99%

“…The group G acts on J 0 = R 2 (x, y) × R 4 (α 0 , α 1 , α 2 , α 3 ) and the action prolongs to J ∞ (R 2 , R 4 ). This action is transitive in 2-jets, and transitive outside the stratum F 3 = 0 in 3-jets, where F 3 is the Liouville relative invariant [29], see also [16]. Differential invariants of this action were counted in [39,44]: h k = 0 for k < 4, h k = 2(k − 1) for k ≥ 4.…”

Section: 12mentioning

confidence: 99%

Poincaré Function for Moduli of Differential-Geometric Structures

Kruglikov¹

2019

MMJ

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The Poincaré function is a compact form of counting moduli in local geometric problems. We discuss its property in relation to V. Arnold's conjecture, and derive this conjecture in the case when the pseudogroup acts algebraically and transitively on the base. Then we survey the known counting results for differential invariants and derive new formulae for several other classification problems in geometry and analysis.

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“…This was initiated by S. Lie and R. Liouville and essentially finished by Tresse [59], see an overview in [27]. In the latter reference the invariants are written via rational generators, and this implies solution of the global equivalence problem (for nonsingular ordinary differential equations of order 2).…”

Section: Examples Of Calculationsmentioning

confidence: 99%