scholarly journals Poincare Metrics on Real Projective Space

1973 ◽  
Vol 23 (1) ◽  
pp. 95-101
Author(s):  
Isaac Chavel
Keyword(s):  
Projective Space ◽  
Real Projective Space

Immersions and Embeddings Up to Cobordism

1971 ◽  
Vol 23 (6) ◽  
pp. 1102-1115 ◽  
Author(s):  
Richard L. W. Brown
Keyword(s):  
Projective Space ◽  
Compact Manifold ◽  
Real Projective Space ◽  
Binary Expansion ◽  
Simple Argument ◽  
Prove Theorem ◽  
Definition Of

In 1944 Whitney proved that any differentiable n-manifold (n ≧ 2) can be (differentiably) immersed in R2n–1[15] and embedded in R2n [14]. Whitney's results are best possible when n = 2r. (One uses a simple argument involving the dual Stiefel-Whitney classes of real projective space Pn. See [9, pp. 14, 15].) However, there is a widely known conjecture that any R-manifold (n ≧ 2) immerses in R2n–α(n) and embeds in R2n–α(n)+1. Here, α(n) denotes the number of ones in the binary expansion of n. We prove (Theorem 5.1) that every compact manifold is cobordant to a manifold that immerses in (2n – α(n))-space and embeds in (2n – α(n) + 1)-space. (See § 1 for the definition of cobordant manifolds.) It is well known that if the conjecture were true it would be the best possible result.

scholarly journals A wall-crossing formula for degrees of Real central projections

2014 ◽  
Vol 25 (04) ◽  
pp. 1450038 ◽  
Author(s):  
Christian Okonek ◽  
Andrei Teleman
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Pole Placement ◽  
Real Projective Space ◽  
Central Projections ◽  
Wall Crossing ◽  
Subspace Problem ◽  
Wall Crossing Formula ◽  
Real Subspace ◽  
Crucial Assumption

The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

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