scholarly journals A measure of non-immersability of the Grassmann manifolds in some euclidean spaces

1998 ◽  
Vol 41 (1) ◽  
pp. 197-205
Author(s):  
Cornel Pintea
Keyword(s):  
Critical Points ◽  
Grassmann Manifold ◽  
Dimensional Vector ◽  
Differentiable Mapping ◽  
Euclidean Spaces ◽  
Grassmann Manifolds ◽  
Vector Subspaces

LetGk, n, be the Grassmann manifold consisting in all non-orientedk-dimensional vector subspaces of the spaceRk+n. In this paper we will show that any differentiable mappingf:Gk, n→Rm, has infinitely many critical points for suitable choices of the numbersm,n,k.

scholarly journals Rational homotopy of maps between certain complex Grassmann manifolds

2018 ◽  
Vol 68 (1) ◽  
pp. 181-196 ◽  
Author(s):  
Prateep Chakraborty ◽  
Shreedevi K. Masuti
Keyword(s):  
Grassmann Manifold ◽  
Dimensional Vector ◽  
Rational Homotopy ◽  
Grassmann Manifolds ◽  
Continuous Map ◽  
Complex Grassmann Manifold ◽  
Vector Subspaces

AbstractLetGn,kdenote the complex Grassmann manifold ofk-dimensional vector subspaces of ℂn. Assumel,k≤ ⌊n/2⌋. We show that, for sufficiently largen, any continuous maph:Gn,l→Gn,kis rationally null homotopic if (i) 1 ≤k<l, (ii) 2 < l <k< 2(l− 1), (iii) 1 < l <k,ldividesnbutldoes not dividek.

scholarly journals Derivations of a Sullivan Model and the Rationalized G -Sequence

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Oteng Maphane
Keyword(s):  
Grassmann Manifold ◽  
Dimensional Vector ◽  
Sullivan Model ◽  
Sullivan Models ◽  
Rational Evaluation ◽  
Vector Subspaces

Let G k , n ℂ for 2 ≤ k < n denote the Grassmann manifold of k -dimensional vector subspaces of ℂ n . In this paper, we compute, in terms of the Sullivan models, the rational evaluation subgroups and, more generally, the G -sequence of the inclusion G 2 , n ℂ ↣ G 2 , n + r ℂ for r ≥ 1 .

scholarly journals Some non-embedding theorems for the Grassmann manifolds G2,n and G3,n

1977 ◽  
Vol 20 (3) ◽  
pp. 177-185 ◽  
Author(s):  
V. Oproiu
Keyword(s):  
Projective Spaces ◽  
Embedding Problem ◽  
Embedding Theorems ◽  
Euclidean Spaces ◽  
Grassmann Manifolds ◽  
Negative Results ◽  
Homotopy Invariants

In recent years, the problem of embedding the projective spaces in Euclidean spaces was studied very much, by different methods. Usually, the negative results on the embedding problem are proved by using suitable homotopy invariants. The best known example of such homotopy invariants is given by the Stiefel–Whitney classes.

Determination of Grassmann Manifolds Which are Boundaries

1991 ◽  
Vol 34 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Parameswaran Sankaran
Keyword(s):  
Grassmann Manifold ◽  
Vector Subspace ◽  
Complex Numbers ◽  
Sufficient Condition ◽  
Grassmann Manifolds

AbstractLetFGn,kdenote the Grassmann manifold of allk-dimensional (left)F-vector subspace ofFnfor F = ℝ, the reals,C, the complex numbers, orHthe quaternions. The problem of determining which of the Grassmannians bound was addressed by the author in [4]. Partial results were obtained in [4] for the caseF= ℝ, including a sufficient condition, due to A. Dold, on n and k for ℝGn,kto bound. Here, we show that Dold's condition is also necessary, and obtain a new proof of sufficiency using the methods of this paper, which cover the complex and quaternionic cases as well.

GEOMETRY OF UNIVERSAL GRASSMANN MANIFOLD FROM ALGEBRAIC POINT OF VIEW

1989 ◽  
Vol 01 (01) ◽  
pp. 1-46 ◽  
Author(s):  
KANEHISA TAKASAKI
Keyword(s):  
Lie Algebra ◽  
Grassmann Manifold ◽  
Point Of View ◽  
Simple Application ◽  
Grassmann Manifolds ◽  
Algebraic Point ◽  
Infinite Dimensional ◽  
Component Theory ◽  
Infinitesimal Transformations ◽  
Algebraic Formulation

An algebraic formulation of the geometry of the universal Grassmann manifold is presented along the line sketched by Sato and Sato [32]. General issues underlying the notion of infinite-dimensional manifolds are also discussed. A particular choice of affine coordinates on Grassmann manifolds, for both the finite- and infinite-dimensional case, turns out to be very useful for the understanding of geometric structures therein. The so-called “Kac-Peterson cocycle”, which is physically a kind of “commutator anomaly”, then arises as a cocycle of a Lie algebra of infinitesimal transformations on the universal Grassmann manifold. The action of group elements for that Lie algebra is also discussed. These ideas are extended to a multi-component theory. A simple application to a non-linear realization of current and Virasoro algebras is presented for illustration.

scholarly journals The expected dimension of a sum of vector subspaces

1992 ◽  
Vol 45 (3) ◽  
pp. 467-477 ◽  
Author(s):  
David E. Dobbs ◽  
Mark J. Lancaster
Keyword(s):  
Vector Space ◽  
Vector Subspace ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Limiting Value ◽  
Vector Subspaces

Let W be an n−dimensional vector space over a field F. It is shown that the expected dimension of a vector subspace of W is n/2. If F is infinite, the expected dimension of a sum of a pair of subspaces of W is (n + 1)/2 if n > 1; and 3/4 if n = 1. If F is finite, with q elements, the expected dimension of a sum of subspaces of W depends on q and n. For fixed n, the limiting value of this expectation as q → ∞ is n if n is even; and n − 1/4 if n is odd. Moreover, if F is finite and n > 1, the expected dimension of a sum of three (not necessarily distinct) subspaces of W has limit n as q → ∞.

RIEMANN DOMAINS OVER BANACH-GRASSMANN MANIFOLDS

2009 ◽  
Vol 02 (03) ◽  
pp. 503-520
Author(s):  
Masaru Nishihara
Keyword(s):  
Banach Space ◽  
Grassmann Manifold ◽  
Complex Banach Space ◽  
Schauder Basis ◽  
Grassmann Manifolds ◽  
Linear Subspaces ◽  
Riemann Domain ◽  
Domain Of Existence ◽  
Complex Linear

Let E be a complex Banach space with a Schauder basis and let G(E; r) be the Grassmann manifold of all r-dimensional complex linear subspaces in E. Let (ω, φ) be a Riemann domain over G(E; r) with ω ≠ G(E; r). Then we show that ω is a domain of existence if and only if ω is pseudoconvex.

Space Forms of Grassmann Manifolds

1963 ◽  
Vol 15 ◽  
pp. 193-205 ◽  
Author(s):  
Joseph A. Wolf
Keyword(s):  
Riemannian Manifolds ◽  
Grassmann Manifold ◽  
Space Form ◽  
Classification Problem ◽  
Grassmann Manifolds ◽  
Space Forms ◽  
Real Complex ◽  
Form Problem ◽  
Spherical Space Form ◽  
Real Euclidean Space

We shall consider the classification problem for space forms of (Riemannian manifolds which are covered by) real, complex, and quaternionic Grassmann manifolds. In the particular case of the real Grassmann manifold of oriented 1-dimensional subspaces of a real Euclidean space, this is the classical "spherical space form problem" of Clifford and Klein. We shall not consider space forms of the Cayley projective plane because it is easy to see that there are no non-trivial ones.

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