scholarly journals On the normal bundle of a homotopy sphere embedded in Euclidean space

Topology ◽  
1965 ◽  
Vol 3 (2) ◽  
pp. 173-181 ◽  
Author(s):  
W.C. Hsiang ◽  
J. Levine ◽  
R.H. Szczarba
Keyword(s):  
Euclidean Space ◽  
Normal Bundle ◽  
Homotopy Sphere

scholarly journals Low codimensional embeddings of Sp(n) and Su(n)

1984 ◽  
Vol 27 (1) ◽  
pp. 25-29 ◽  
Author(s):  
G. Walker ◽  
R. M. W. Wood
Keyword(s):  
Euclidean Space ◽  
Lie Group ◽  
Maximal Torus ◽  
Unitary Group ◽  
Normal Bundle ◽  
Symplectic Group ◽  
Adjoint Representation ◽  
General Technique ◽  
Special Cases ◽  
Connected Lie Group

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

On Immersion of Manifolds

1960 ◽  
Vol 12 ◽  
pp. 529-534 ◽  
Author(s):  
Hans Samelson
Keyword(s):  
Euclidean Space ◽  
Tangent Bundle ◽  
Normal Bundle ◽  
Vector Fields ◽  
Elementary Proof ◽  
Homology Class ◽  
Oriented Manifold ◽  
Normal Degree ◽  
Special Case

In (3) R. Lashof and S. Smale proved among other things the following theorem. If the compact oriented manifold M is immersed into the oriented manifold M', with dim M' ≥ dim M + 2, then the normal degree of the immersion is equal to the Euler-Poincaré characteristic x of M reduced module the characteristic x’ of M'. If M’ is not compact, x' is replaced by 0. “Manifold” always means C∞-manifold. An immersion is a differentiable (that is, C∞) map f whose differential df is non-singular throughout. The normal degree is defined in a certain fashion using the normal bundle of M in M', derived from f, and injecting it into the tangent bundle of M'It is our purpose to give an elementary proof, using vector fields, of this theorem, and at the same time to identify the homology class that represents the normal degree (Theorem I), and to give a proof, using the theory of Morse, for the special case M’ = Euclidean space (Theorem II).

Submanifolds with Nonparallel First Normal Bundle

1994 ◽  
Vol 37 (3) ◽  
pp. 330-337 ◽  
Author(s):  
Marcos Dajczer ◽  
Ruy Tojeiro
Keyword(s):  
Euclidean Space ◽  
Sectional Curvature ◽  
Normal Bundle ◽  
Second Fundamental Form ◽  
Low Rank ◽  
Geometric Description ◽  
Normal Connection ◽  
Flat Normal Bundle ◽  
The Second Fundamental Form ◽  
Flat Submanifolds

AbstractWe provide a complete local geometric description of submanifolds of spaces with constant sectional curvature where the first normal spaces, that is, the subspaces spanned by the second fundamental form, form a vector subbundle of the normal bundle of low rank which is nonparallel in the normal connection. We also characterize flat submanifolds with flat normal bundle in Euclidean space satisfying the helix property.

scholarly journals A cohomological formula for the Atiyah–Patodi–Singer index on manifolds with boundary

2014 ◽  
Vol 06 (01) ◽  
pp. 27-74 ◽  
Author(s):  
P. Carrillo Rouse ◽  
J. M. Lescure ◽  
B. Monthubert
Keyword(s):  
Boundary Condition ◽  
Euclidean Space ◽  
Algebraic Topology ◽  
Normal Bundle ◽  
Pseudodifferential Operators ◽  
Index Theorem ◽  
Manifolds With Boundary ◽  
C Algebras ◽  
Manifold With Boundary ◽  
K Theory

The main result of this paper is a new Atiyah–Singer type cohomological formula for the index of Fredholm pseudodifferential operators on a manifold with boundary. The nonlocality of the chosen boundary condition prevents us to apply directly the methods used by Atiyah and Singer in [4, 5]. However, by using the K-theory of C*-algebras associated to some groupoids, which generalizes the classical K-theory of spaces, we are able to understand the computation of the APS index using classic algebraic topology methods (K-theory and cohomology). As in the classic case of Atiyah–Singer ([4, 5]), we use an embedding into a Euclidean space to express the index as the integral of a true form on a true space, the integral being over a C∞-manifold called the singular normal bundle associated to the embedding. Our formula is based on a K-theoretical Atiyah–Patodi–Singer theorem for manifolds with boundary that is inspired by Connes' tangent groupoid approach, it is not a groupoid interpretation of the famous Atiyah–Patodi–Singer index theorem.

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.

ANTI-SELF-DUAL SOLUTIONS OF THE YANG-MILLS EQUATIONS IN 4n DIMENSIONS

1992 ◽  
Vol 07 (23) ◽  
pp. 2077-2085 ◽  
Author(s):  
A. D. POPOV
Keyword(s):  
Scalar Field ◽  
Euclidean Space ◽  
Lie Algebra ◽  
Linear Equations ◽  
Dual Solutions ◽  
Gauge Fields ◽  
System Of Equations ◽  
Semisimple Lie Algebra ◽  
Yang Mills ◽  
Self Duality

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.

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