Surfaces of codimension two with zero normal torsion carrying a conjugate coordinate net in Euclidean space

1989 ◽  
Vol 30 (1) ◽  
pp. 128-136 ◽  
Author(s):  
V. T. Fomenko ◽  
A. N. Zubkov
Keyword(s):  
Euclidean Space ◽  
Codimension Two ◽  
Normal Torsion

On submanifolds with zero normal torsion in Euclidean space

1994 ◽  
Vol 35 (3) ◽  
pp. 470-478
Author(s):  
I. I. Bodrenko
Keyword(s):  
Euclidean Space ◽  
Normal Torsion

Unknotting tori in codimension one and spheres in codimension two

1965 ◽  
Vol 61 (3) ◽  
pp. 659-664 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Piecewise Linear ◽  
Differential Topology ◽  
Standard Unit ◽  
Codimension Two ◽  
Codimension One

We shall present this paper in the framework and terminology of differential topology though all our arguments are valid in the piecewise linear ease also, under local un-knottedness hypotheses. In particular we use Rp for Euclidean space of dimension p, Sp−1 for the standard unit sphere in it, and Dp for the disc which it bounds.

Surfaces in euclidean space with planar normal connectivity and zero normal torsion

1993 ◽  
Vol 54 (1) ◽  
pp. 667-676
Author(s):  
A. N. Zubkov ◽  
V. T. Fomenko
Keyword(s):  
Euclidean Space ◽  
Normal Torsion ◽  
Surfaces In Euclidean Space

Compactification of Submanifolds in Euclidean Space by the Inversion

2018 ◽  
Author(s):  
Kazuyuki Enomoto
Keyword(s):  
Euclidean Space

Symplectic capacities

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Euclidean Space ◽  
Generating Functions ◽  
Weinstein Conjecture ◽  
Finite Dimensional ◽  
Hofer Metric

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

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