Insight in the Rumin Cohomology and Orientability Properties of the Heisenberg Group

Forskningsoutput: AvhandlingLicentiatavhandling

Sammanfattning

The purpose of this study is to analyse two related topics: the Rumin cohomology and the H-orientability in the Heisenberg group H^n. In the first three chapters we carefully describe the Rumin cohomology with particular emphasis at the second order differential operator D, giving examples in the cases n+1 and n+2. We also show the commutation between all Rumin differential operators and the pullback by a contact map and, more generally, describe pushforward and pullback explicitly in different situations. Differential forms can be used to define the notion of orientability; indeed in the fourth chapter we define the H-orientability for H-regular surfaces and we prove that H-orientability implies standard orientability, while the opposite is not always true. Finally we show that, up to one point, a Mobius strip in H^1 is a H-regular surface and we use this fact to prove that there exist H-regular non-H-orientable surfaces, at least in the case n+1. This opens the possibility for an analysis of Heisenberg currents mod 2.
Originalspråkengelska
Tilldelande institution
  • Helsingfors universitet
Handledare
  • Holopainen, Ilkka, Handledare
Tilldelningsdatum6 nov 2018
Förlag
StatusPublicerad - 8 nov 2018
MoE-publikationstypG3 Licentiatavhandling

Vetenskapsgrenar

  • 111 Matematik

Citera det här

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title = "Insight in the Rumin Cohomology and Orientability Properties of the Heisenberg Group",
abstract = "The purpose of this study is to analyse two related topics: the Rumin cohomology and the H-orientability in the Heisenberg group H^n. In the first three chapters we carefully describe the Rumin cohomology with particular emphasis at the second order differential operator D, giving examples in the cases n+1 and n+2. We also show the commutation between all Rumin differential operators and the pullback by a contact map and, more generally, describe pushforward and pullback explicitly in different situations. Differential forms can be used to define the notion of orientability; indeed in the fourth chapter we define the H-orientability for H-regular surfaces and we prove that H-orientability implies standard orientability, while the opposite is not always true. Finally we show that, up to one point, a Mobius strip in H^1 is a H-regular surface and we use this fact to prove that there exist H-regular non-H-orientable surfaces, at least in the case n+1. This opens the possibility for an analysis of Heisenberg currents mod 2.",
keywords = "111 Mathematics, Heisenberg group, sub-Riemannian geometry, Differential geometry, Rumin cohomology, Heisenberg-orientability",
author = "Giovanni Canarecci",
year = "2018",
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language = "English",
publisher = "University of Helsinki",
address = "Finland",
school = "University of Helsinki",

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Insight in the Rumin Cohomology and Orientability Properties of the Heisenberg Group. / Canarecci, Giovanni.

University of Helsinki, 2018. 111 s.

Forskningsoutput: AvhandlingLicentiatavhandling

TY - THES

T1 - Insight in the Rumin Cohomology and Orientability Properties of the Heisenberg Group

AU - Canarecci, Giovanni

PY - 2018/11/8

Y1 - 2018/11/8

N2 - The purpose of this study is to analyse two related topics: the Rumin cohomology and the H-orientability in the Heisenberg group H^n. In the first three chapters we carefully describe the Rumin cohomology with particular emphasis at the second order differential operator D, giving examples in the cases n+1 and n+2. We also show the commutation between all Rumin differential operators and the pullback by a contact map and, more generally, describe pushforward and pullback explicitly in different situations. Differential forms can be used to define the notion of orientability; indeed in the fourth chapter we define the H-orientability for H-regular surfaces and we prove that H-orientability implies standard orientability, while the opposite is not always true. Finally we show that, up to one point, a Mobius strip in H^1 is a H-regular surface and we use this fact to prove that there exist H-regular non-H-orientable surfaces, at least in the case n+1. This opens the possibility for an analysis of Heisenberg currents mod 2.

AB - The purpose of this study is to analyse two related topics: the Rumin cohomology and the H-orientability in the Heisenberg group H^n. In the first three chapters we carefully describe the Rumin cohomology with particular emphasis at the second order differential operator D, giving examples in the cases n+1 and n+2. We also show the commutation between all Rumin differential operators and the pullback by a contact map and, more generally, describe pushforward and pullback explicitly in different situations. Differential forms can be used to define the notion of orientability; indeed in the fourth chapter we define the H-orientability for H-regular surfaces and we prove that H-orientability implies standard orientability, while the opposite is not always true. Finally we show that, up to one point, a Mobius strip in H^1 is a H-regular surface and we use this fact to prove that there exist H-regular non-H-orientable surfaces, at least in the case n+1. This opens the possibility for an analysis of Heisenberg currents mod 2.

KW - 111 Mathematics

KW - Heisenberg group

KW - sub-Riemannian geometry

KW - Differential geometry

KW - Rumin cohomology

KW - Heisenberg-orientability

M3 - Licenciate's thesis

PB - University of Helsinki

ER -