Local symplectic invariants for curves

Niky Kamran; Peter Olver; Keti Tenenblat

doi:10.1142/S0219199709003326

Local symplectic invariants for curves

Niky Kamran, Peter Olver, Keti Tenenblat

School of Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

We consider curves in ℝ²ⁿ endowed with the standard symplectic structure. We introduce the concept of symplectic arc length for curves. We construct an adapted symplectic Frenet frame and we define 2n - 1 local differential invariants that we call symplectic curvatures of the curve. We prove that up to a rigid symplectic motion of ℝ²ⁿ, there exists a unique curve with prescribed symplectic curvatures. We characterize the curves in ℝ⁴ with constant symplectic curvatures.

Original language	English (US)
Pages (from-to)	165-183
Number of pages	19
Journal	Communications in Contemporary Mathematics
Volume	11
Issue number	2
DOIs	https://doi.org/10.1142/S0219199709003326
State	Published - Apr 2009

Bibliographical note

Funding Information:
The research of the first author was supported in part by NSERC grant RGPIN 105490-2004, the second author by NSF Grant DMS 05-05293, and the third author by CNPq Grant 306117/2006-1.

Keywords

Moving frames
Symplectic curvatures
Symplectic curves

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10.1142/S0219199709003326

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title = "Local symplectic invariants for curves",

abstract = "We consider curves in ℝ2n endowed with the standard symplectic structure. We introduce the concept of symplectic arc length for curves. We construct an adapted symplectic Frenet frame and we define 2n - 1 local differential invariants that we call symplectic curvatures of the curve. We prove that up to a rigid symplectic motion of ℝ2n, there exists a unique curve with prescribed symplectic curvatures. We characterize the curves in ℝ4 with constant symplectic curvatures.",

keywords = "Moving frames, Symplectic curvatures, Symplectic curves",

author = "Niky Kamran and Peter Olver and Keti Tenenblat",

note = "Funding Information: The research of the first author was supported in part by NSERC grant RGPIN 105490-2004, the second author by NSF Grant DMS 05-05293, and the third author by CNPq Grant 306117/2006-1.",

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AU - Kamran, Niky

AU - Olver, Peter

AU - Tenenblat, Keti

N1 - Funding Information: The research of the first author was supported in part by NSERC grant RGPIN 105490-2004, the second author by NSF Grant DMS 05-05293, and the third author by CNPq Grant 306117/2006-1.

PY - 2009/4

Y1 - 2009/4

N2 - We consider curves in ℝ2n endowed with the standard symplectic structure. We introduce the concept of symplectic arc length for curves. We construct an adapted symplectic Frenet frame and we define 2n - 1 local differential invariants that we call symplectic curvatures of the curve. We prove that up to a rigid symplectic motion of ℝ2n, there exists a unique curve with prescribed symplectic curvatures. We characterize the curves in ℝ4 with constant symplectic curvatures.

AB - We consider curves in ℝ2n endowed with the standard symplectic structure. We introduce the concept of symplectic arc length for curves. We construct an adapted symplectic Frenet frame and we define 2n - 1 local differential invariants that we call symplectic curvatures of the curve. We prove that up to a rigid symplectic motion of ℝ2n, there exists a unique curve with prescribed symplectic curvatures. We characterize the curves in ℝ4 with constant symplectic curvatures.

KW - Moving frames

KW - Symplectic curvatures

KW - Symplectic curves

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VL - 11

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EP - 183

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

IS - 2

ER -

Local symplectic invariants for curves

Abstract

Bibliographical note

Keywords

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