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Geometry of Hörmander’s Vector Fields

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An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

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Abstract

The chapter focusses of some basic ideas in the geometry of Hörmander’s vector fields. We start with a discussion of the basic concept of connectivity, and some of its physical meanings which have been known for a long time. We then pass to survey some ideas and fundamental results about the metric induced by a system of vector fields, which are contained in some papers of the middle 1980’s. This area is by a now an independent field of study, but here we are mainly interested in its relevance in connection with the study of second order PDEs of Hörmander’s type.

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Notes

  1. 1.

    The original paper by Carathéodory is [7], in German. An English translation can be found in the book [25, Chap. 12]. To the reader who is interested in Carathéodory’s ideas on thermodynamics, however, I suggest the reading of some parts of an essay written by Max Born on that subject, see [4, Chap. V and Appendix 6, 7].

  2. 2.

    Recall that one of Hilbert’s problems posed in the international congress of mathematics held in Paris in 1900 consisted in giving an axiomatization of physical theories, following the general trend to axiomatization which pervaded mathematics in those years.

  3. 3.

    Here we follow the simplified presentation of the theory given in [4, Chap. 5]; Carathéodory’s approach is more general and abstract, and involves Pfaffian forms of \(n\) variables.

  4. 4.

    For an overview of the subject I particularly suggest the reading of Jurdjevic’ introduction to his book and the introduction to Chap. 1.

  5. 5.

    A clear example of such control system is rowing a boat on a river: the river’s drift is out of our control.

  6. 6.

    Actually, Gromov in [19, p. 87] suggests that some form of the connectivity theorem could be known to Lagrange in the context of nonholonomic mechanics.

  7. 7.

    The reader will recognize some analogy with the controllability problem for an affine control system with drift, discussed in the previous paragraph.

  8. 8.

    Actually the authors prove the equivalence between five different distances, but here we will concentrate just on two of them.

  9. 9.

    The first use of this term appeared in Gromov’ book of 1981 “Structures métriques pour les variétés riemanniennes” (edited by J. Lafontaine and P. Pansu), later transformed in [20].

  10. 10.

    Gromov refers to the 1981 edition of [20], see the previous footnote.

  11. 11.

    The following bibliographical account is quoted from [5, p. 3].

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    Marco Bramanti

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Bramanti, M. (2014). Geometry of Hörmander’s Vector Fields. In: An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-02087-7_4

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