Nonlinear Grassmannians: plain, decorated, augmented

Cornelia Vizman; Stefan Haller

arxiv: 2503.15363 · v1 · submitted 2025-03-19 · 🧮 math.DG · math.SG

Nonlinear Grassmannians: plain, decorated, augmented

Stefan Haller , Cornelia Vizman This is my paper

Pith reviewed 2026-05-23 00:31 UTC · model grok-4.3

classification 🧮 math.DG math.SG

keywords nonlinear Grassmannianscoadjoint orbitsdiffeomorphism groupssymplectic Fréchet manifoldsdecoration functorsaugmentation functorsinfinite-dimensional symplectic geometry

0 comments

The pith

Decorated and augmented nonlinear Grassmannians parametrize coadjoint orbits of diffeomorphism groups as smooth symplectic Fréchet manifolds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out a general framework of decoration and augmentation functors that turn nonlinear Grassmannians into spaces carrying smooth Fréchet manifold structures. These structures are compatible with the coadjoint action of classical diffeomorphism groups, so the orbits themselves become smooth symplectic Fréchet manifolds. The construction does not produce new orbits; instead it supplies a single method that works uniformly for the known families. A reader would care because infinite-dimensional symplectic geometry has often required separate arguments for each group, and a uniform Grassmannian model simplifies both proofs and comparisons.

Core claim

Decorated and augmented nonlinear Grassmannians can be used to parametrize coadjoint orbits of classical diffeomorphism groups. The authors supply decoration and augmentation functors that induce smooth Fréchet manifold structures on these Grassmannians in a way compatible with the coadjoint action, thereby equipping the orbits with the structure of smooth symplectic Fréchet manifolds. The resulting description is uniform across the orbits that arise this way.

What carries the argument

Decoration and augmentation functors on nonlinear Grassmannians, which produce spaces that inherit a smooth Fréchet manifold structure compatible with the coadjoint action and the symplectic form.

If this is right

The coadjoint orbits acquire a well-defined smooth manifold topology and differentiable structure from the underlying Grassmannian.
The Kirillov-Kostant-Souriau symplectic form on each orbit descends from the geometry of the decorated or augmented Grassmannian.
The same functorial construction applies without modification to the coadjoint orbits of the diffeomorphism group of a manifold, its volume-preserving subgroup, and related classical groups.
Local charts and transition maps on the orbits are inherited directly from the Grassmannian model rather than constructed ad hoc.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The functorial approach may extend to coadjoint orbits of other infinite-dimensional Lie groups whose actions preserve additional geometric structures.
Explicit coordinate computations or curvature calculations on these orbits could be carried out by lifting them to the Grassmannian level where local models are simpler.
The uniform description might make it easier to compare orbit geometry across different base manifolds or different choices of decoration.

Load-bearing premise

The decoration and augmentation functors can be defined so the resulting spaces carry a smooth Fréchet manifold structure compatible with the coadjoint action and symplectic form.

What would settle it

An explicit example of a classical diffeomorphism group and orbit where no choice of decoration or augmentation yields a Fréchet manifold structure whose tangent spaces support a nondegenerate closed two-form invariant under the group action.

read the original abstract

Decorated and augmented nonlinear Grassmannians can be used to parametrize coadjoint orbits of classical diffeomorphism groups. We provide a general framework for decoration and augmentation functors that facilitates the construction of a smooth structure on decorated or augmented nonlinear Grassmannians. This permits to equip the corresponding coadjoint orbits with the structure of a smooth symplectic Frechet manifold. The coadjoint orbits obtained in this way are not new. Here, we provide a uniform description of their smooth structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies a general functorial framework for decoration and augmentation that gives a uniform smooth Fréchet structure on already-known coadjoint orbits of diffeomorphism groups.

read the letter

The central new piece is the general construction of decoration and augmentation functors on nonlinear Grassmannians. These functors are set up so the resulting spaces carry a smooth Fréchet atlas, the coadjoint action stays smooth, and the Kirillov-Kostant-Souriau form is well-defined. The orbits themselves are explicitly not claimed as new; the contribution is the uniform re-description. That framing is honest and keeps the scope clear. The functorial approach looks like a practical way to organize several known examples under one construction, which can save time when checking smoothness and compatibility in infinite-dimensional symplectic geometry. The abstract states the functors are built precisely to inherit the required properties, and the stress-test finds no internal contradiction in that claim. On the soft side, the provided abstract gives no derivation or proof sketch for why the functors preserve the Fréchet structure and the symplectic form, so the load-bearing verification step cannot be checked from the text alone. If the full paper supplies explicit definitions and checks for the classical diffeomorphism groups, that would close the gap; otherwise the claim rests on the construction details. The work sits squarely inside infinite-dimensional differential geometry and is aimed at readers who already work with coadjoint orbits of Diff(M) or related groups. Those readers can extract a reusable technique even if they already know the individual orbits. It is coherent on its own terms and shows clear engagement with the literature on these spaces. I would send it to a serious referee rather than desk-reject, because the uniform framework is a concrete organizational advance worth checking in detail.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces a general framework for decoration and augmentation functors on nonlinear Grassmannians. It claims that the resulting decorated and augmented spaces furnish a uniform parametrization of known coadjoint orbits of classical diffeomorphism groups, and that the functors can be defined so the spaces carry a smooth Fréchet manifold structure compatible with the coadjoint action and the Kirillov-Kostant-Souriau symplectic form. The paper explicitly states that the orbits are not new and positions its contribution as providing this uniform description of their smooth structures.

Significance. If the functorial constructions are carried out rigorously and the smoothness and compatibility properties are verified, the work would supply a systematic, uniform language for describing the infinite-dimensional geometry of these coadjoint orbits. This could streamline arguments that currently rely on case-by-case constructions for different diffeomorphism groups. The manuscript does not claim novelty for the orbits themselves, which limits the scope of the advance to a re-description.

major comments (2)

[Abstract] Abstract: the assertion that the decoration and augmentation functors 'facilitate the construction of a smooth structure' and permit the coadjoint orbits to be equipped with a smooth symplectic Fréchet manifold structure is stated without any derivation, explicit definition of the functors, or verification that the resulting atlas is Fréchet, that the coadjoint action remains smooth, or that the KKS form is well-defined. This verification is the load-bearing step for the central claim.
[Abstract] The manuscript frames the contribution as functorial and general, yet provides no indication of how the functors are constructed on the underlying nonlinear Grassmannians or why they automatically inherit the required Fréchet atlas from the plain case. Without this, the claim that the resulting spaces carry a smooth structure compatible with the coadjoint action cannot be assessed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The major comments concern the level of detail in the abstract regarding the functor constructions and smoothness verifications. We address each point below and will revise the abstract accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that the decoration and augmentation functors 'facilitate the construction of a smooth structure' and permit the coadjoint orbits to be equipped with a smooth symplectic Fréchet manifold structure is stated without any derivation, explicit definition of the functors, or verification that the resulting atlas is Fréchet, that the coadjoint action remains smooth, or that the KKS form is well-defined. This verification is the load-bearing step for the central claim.

Authors: We agree the abstract is a high-level summary and does not contain the derivations. The explicit definitions of the decoration and augmentation functors appear in Sections 3 and 4, where they are constructed on the underlying nonlinear Grassmannians. The verification that the resulting spaces carry a Fréchet atlas, that the coadjoint action is smooth, and that the KKS form is well-defined is carried out in Section 5. We will revise the abstract to add a brief clause indicating that these properties are established in the main text. revision: yes
Referee: [Abstract] The manuscript frames the contribution as functorial and general, yet provides no indication of how the functors are constructed on the underlying nonlinear Grassmannians or why they automatically inherit the required Fréchet atlas from the plain case. Without this, the claim that the resulting spaces carry a smooth structure compatible with the coadjoint action cannot be assessed.

Authors: The paper develops the general framework precisely to show how the functors are defined on nonlinear Grassmannians so that the Fréchet atlas is inherited from the plain case; this inheritance and the compatibility with the coadjoint action are proved in the body. We will revise the abstract to include a short phrase noting that the functors are constructed to preserve the smooth structure, directing readers to the relevant sections for the details. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction of functors on standard objects

full rationale

The paper explicitly states that the coadjoint orbits are not new and frames its contribution as providing a uniform description via decoration and augmentation functors on nonlinear Grassmannians. The abstract and reader's summary indicate a direct construction of smooth Fréchet structures compatible with the coadjoint action and KKS form, without any indication of self-referential definitions, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs. The derivation chain is self-contained as a functorial re-parametrization of known objects.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the contribution is a functorial construction in differential geometry.

pith-pipeline@v0.9.0 · 5597 in / 1149 out tokens · 97544 ms · 2026-05-23T00:31:39.178425+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Decorated and augmented nonlinear Grassmannians can be used to parametrize coadjoint orbits of classical diffeomorphism groups... uniform description of their smooth structures.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A decoration functor D is a functor from the category of closed manifolds... to Fréchet manifolds such that the Diff(S) action on D(S) is smooth.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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Binz and H.R

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Gay-Balmaz and C

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Gay-Balmaz and C

F. Gay-Balmaz and C. Vizman, Principal bundles of embeddings , Ann. Global Anal. Geom. 46 (2014), 293–312

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Gay-Balmaz and C

F. Gay-Balmaz and C. Vizman, Isotropic submanifolds and coadjoint orbits of the Hamilto nian group, J. Symp. Geom. 17 (2019), 663–702

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Gay-Balmaz and C

F. Gay-Balmaz and C. Vizman, Coadjoint orbits of vortex sheets in ideal ﬂuids , J. Geom. Phys. 197 (2024), Paper No. 105096, 13 pp

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Gl¨ ockner and K.-H

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G. A. Goldin, R. Menikoﬀ, and D. H. Sharp, Quantum vortex conﬁgurations in three dimensions. Phys. Rev. Lett. 67 (1991), 3499–3502

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Haller and C

S. Haller and C. Vizman, Nonlinear Grassmannians as coadjoint orbits , Math. Ann. 329 (2004), 771–785

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Haller and C

S. Haller and C. Vizman, Nonlinear ﬂag manifolds as coadjoint orbits , Ann. Global Anal. Geom. 58 (2020), 385–413

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Haller and C

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Haller and C

S. Haller and C. Vizman, Weighted nonlinear ﬂag manifolds as coadjoint orbits , Canad. J. Math. 76 (2024), 1664–1694

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Haller and C

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D. D. Holm and J. E. Marsden, Moment maps and measure-valued solutions (peakons, ﬁlamen ts, and sheets) for the EPDiﬀ equation. In The breadth of symplectic and Poisson geometry , volume 232 of Progr. Math., pages 203–235. Birkh¨ auser Boston, Boston, MA, 2005. 31

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R. S. Ismagilov, Representations of Inﬁnite-Dimensional Groups. Translated from the Russian Manuscript by D. Deart., Translations of Mathematical Monographs, vol. 152. American Mathematical Society, Providence (1996)

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Khesin, Symplectic structures and dynamics on vortex membranes , Moscow Math

B. Khesin, Symplectic structures and dynamics on vortex membranes , Moscow Math. J. 12 (2013), 413–434

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Kriegl and P

A. Kriegl and P. W. Michor. The convenient setting of global analysis. Mathematical Surveys and Monographs

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J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for inco mpressible ﬂuids, Phys. D, 7(1983), 305–323

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P. W. Michor, Manifolds of smooth maps. III. The princip al bundle of embeddings of a noncompact smooth manifold. Cahiers Topologie G´ eom. Diﬀ´ erentielle21, 325–337 (1980)

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P. W. Michor, Manifolds of diﬀerentiable mappings. Shi va Mathematics Series 3. Shiva Publishing Ltd., Nantwich (1980)

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[26]

P. W. Michor, Topics in diﬀerential geometry. Grad. Stud. Math. 93. American Mathematical Society, Provi- dence, RI, 2008

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[27]

Moser, On the volume elements on a manifold , Trans

J. Moser, On the volume elements on a manifold , Trans. Amer. Math. Soc. 120 (1965), 286–294

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[28]

Milnor, Remarks on inﬁnite-dimensional Lie groups , pp

J. Milnor, Remarks on inﬁnite-dimensional Lie groups , pp. 1007–1057 in: B. DeWitt and R. Stora (eds), Rela- tivit´ e, groupes et topologie II (Les Houches, 1983), North Holland, Amsterdam, 1984

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[29]

I. M. Singer and S. Sternberg. The inﬁnite groups of Lie and Cartan. I. The transitive group s. J. Analyse Math., 15:1-114, 1965

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[30]

Weinstein, Lectures on symplectic manifolds

A. Weinstein, Lectures on symplectic manifolds. Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8–12, 1976. Regio nal Conference Series in Mathematics, No. 29. American Mathematical Society, Providence, R.I., 1977

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Weinstein, Neighborhood classiﬁcation of isotropic embeddings , J

A. Weinstein, Neighborhood classiﬁcation of isotropic embeddings , J. Diﬀ. Geom. 16 (1981), 125–128

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Weinstein, The local structure of Poisson manifolds , J

A. Weinstein, The local structure of Poisson manifolds , J. Diﬀerential Geom., 18 (1983), 523–557

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[33]

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A. Weinstein, Connections of Berry and Hannay type for moving Lagrangian s ubmanifolds, Adv. Math., 82 (1990), 133–159. Stefan Haller, Department of Mathematics, University of Vi enna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria. ORCID 0000-0002-7064-2215 Email address : stefan.haller@univie.ac.at Cornelia Vizman, Department of Mathematics, West Unive...

work page 1990

[1] [1]

M. C. Abbati, R. Cirellu, A. Mania and P. Michor, The Lie group of automorphisms of a principle bundle , Journal of Geometry and Physics, 6 (1989), 215–235

work page 1989

[2] [2]

Binz and H.R

E. Binz and H.R. Fischer, The manifold of embeddings of a c losed manifold. With an appendix by P. Michor. Lecture Notes in Phys. 139, Diﬀerential geometric methods in mathematical physics (P roc. Internat. Conf., Tech. Univ. Clausthal, Clausthal-Zellerfeld, 1978), pp. 3 10–329, Springer, Berlin-New York (1981)

work page 1978

[3] [3]

E. Cartan. Les groupes de transformations continus, inﬁnis, simples . Ann. Sci. ´Ecole Norm. Sup. (3), 26:93-161, 1909

work page 1909

[4] [4]

Gay-Balmaz and C

F. Gay-Balmaz and C. Vizman, Dual pairs in ﬂuid dynamics , Ann. Global Anal. Geom. 41 (2012), 1–24

work page 2012

[5] [5]

Gay-Balmaz and C

F. Gay-Balmaz and C. Vizman, Principal bundles of embeddings , Ann. Global Anal. Geom. 46 (2014), 293–312

work page 2014

[6] [6]

Gay-Balmaz and C

F. Gay-Balmaz and C. Vizman, Isotropic submanifolds and coadjoint orbits of the Hamilto nian group, J. Symp. Geom. 17 (2019), 663–702

work page 2019

[7] [7]

Gay-Balmaz and C

F. Gay-Balmaz and C. Vizman, Coadjoint orbits of vortex sheets in ideal ﬂuids , J. Geom. Phys. 197 (2024), Paper No. 105096, 13 pp

work page 2024

[8] [8]

Gl¨ ockner and K.-H

H. Gl¨ ockner and K.-H. Neeb, Inﬁnite-Dimensional Lie Groups , Book in preparation

work page

[9] [9]

G. A. Goldin, R. Menikoﬀ, and D. H. Sharp, Diﬀeomorphism groups and quantized vortex ﬁlaments , Phys. Rev. Lett. 58 (1987), 2162–2164

work page 1987

[10] [10]

G. A. Goldin, R. Menikoﬀ, and D. H. Sharp, Quantum vortex conﬁgurations in three dimensions. Phys. Rev. Lett. 67 (1991), 3499–3502

work page 1991

[11] [11]

Haller and C

S. Haller and C. Vizman, Nonlinear Grassmannians as coadjoint orbits , Math. Ann. 329 (2004), 771–785

work page 2004

[12] [12]

Haller and C

S. Haller and C. Vizman, Nonlinear ﬂag manifolds as coadjoint orbits , Ann. Global Anal. Geom. 58 (2020), 385–413

work page 2020

[13] [13]

Haller and C

S. Haller and C. Vizman, A dual pair for the contact group , Math. Z. 301 (2022), 2937–2973

work page 2022

[14] [14]

Haller and C

S. Haller and C. Vizman, Weighted nonlinear ﬂag manifolds as coadjoint orbits , Canad. J. Math. 76 (2024), 1664–1694

work page 2024

[15] [15]

Haller and C

S. Haller and C. Vizman, A dual pair for the volume preserving diﬀeomorphism group , arXiv:2405.10737 (2024)

work page arXiv 2024

[16] [16]

R. S. Hamilton, The inverse function theorem of Nash and Moser , Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65–222

work page 1982

[17] [17]

D. D. Holm and J. E. Marsden, Moment maps and measure-valued solutions (peakons, ﬁlamen ts, and sheets) for the EPDiﬀ equation. In The breadth of symplectic and Poisson geometry , volume 232 of Progr. Math., pages 203–235. Birkh¨ auser Boston, Boston, MA, 2005. 31

work page 2005

[18] [18]

R. S. Ismagilov, Representations of Inﬁnite-Dimensional Groups. Translated from the Russian Manuscript by D. Deart., Translations of Mathematical Monographs, vol. 152. American Mathematical Society, Providence (1996)

work page 1996

[19] [19]

Khesin, Symplectic structures and dynamics on vortex membranes , Moscow Math

B. Khesin, Symplectic structures and dynamics on vortex membranes , Moscow Math. J. 12 (2013), 413–434

work page 2013

[20] [20]

Kriegl and P

A. Kriegl and P. W. Michor. The convenient setting of global analysis. Mathematical Surveys and Monographs

work page

[21] [21]

American Mathematical Society, Providence, RI, 1997

work page 1997

[22] [22]

Lee, Geometric structures on spaces of weighted submanifolds , SIGMA, 5(2009), Article No

B. Lee, Geometric structures on spaces of weighted submanifolds , SIGMA, 5(2009), Article No. 099, 49pp

work page 2009

[23] [23]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for inco mpressible ﬂuids, Phys. D, 7(1983), 305–323

work page 1983

[24] [24]

P. W. Michor, Manifolds of smooth maps. III. The princip al bundle of embeddings of a noncompact smooth manifold. Cahiers Topologie G´ eom. Diﬀ´ erentielle21, 325–337 (1980)

work page 1980

[25] [25]

P. W. Michor, Manifolds of diﬀerentiable mappings. Shi va Mathematics Series 3. Shiva Publishing Ltd., Nantwich (1980)

work page 1980

[26] [26]

P. W. Michor, Topics in diﬀerential geometry. Grad. Stud. Math. 93. American Mathematical Society, Provi- dence, RI, 2008

work page 2008

[27] [27]

Moser, On the volume elements on a manifold , Trans

J. Moser, On the volume elements on a manifold , Trans. Amer. Math. Soc. 120 (1965), 286–294

work page 1965

[28] [28]

Milnor, Remarks on inﬁnite-dimensional Lie groups , pp

J. Milnor, Remarks on inﬁnite-dimensional Lie groups , pp. 1007–1057 in: B. DeWitt and R. Stora (eds), Rela- tivit´ e, groupes et topologie II (Les Houches, 1983), North Holland, Amsterdam, 1984

work page 1983

[29] [29]

I. M. Singer and S. Sternberg. The inﬁnite groups of Lie and Cartan. I. The transitive group s. J. Analyse Math., 15:1-114, 1965

work page 1965

[30] [30]

Weinstein, Lectures on symplectic manifolds

A. Weinstein, Lectures on symplectic manifolds. Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8–12, 1976. Regio nal Conference Series in Mathematics, No. 29. American Mathematical Society, Providence, R.I., 1977

work page 1976

[31] [31]

Weinstein, Neighborhood classiﬁcation of isotropic embeddings , J

A. Weinstein, Neighborhood classiﬁcation of isotropic embeddings , J. Diﬀ. Geom. 16 (1981), 125–128

work page 1981

[32] [32]

Weinstein, The local structure of Poisson manifolds , J

A. Weinstein, The local structure of Poisson manifolds , J. Diﬀerential Geom., 18 (1983), 523–557

work page 1983

[33] [33]

Weinstein, Connections of Berry and Hannay type for moving Lagrangian s ubmanifolds, Adv

A. Weinstein, Connections of Berry and Hannay type for moving Lagrangian s ubmanifolds, Adv. Math., 82 (1990), 133–159. Stefan Haller, Department of Mathematics, University of Vi enna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria. ORCID 0000-0002-7064-2215 Email address : stefan.haller@univie.ac.at Cornelia Vizman, Department of Mathematics, West Unive...

work page 1990