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Skew Flat Fibrations and Totally Con...

Harrison, Michael A.

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Skew Flat Fibrations and Totally Convex Immersions.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Skew Flat Fibrations and Totally Convex Immersions./
Author:	Harrison, Michael A.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2017,
Description:	45 p.
Notes:	Source: Dissertation Abstracts International, Volume: 79-04(E), Section: B.
Contained By:	Dissertation Abstracts International79-04B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10666512
ISBN:	9780355330540

Skew Flat Fibrations and Totally Convex Immersions.
Harrison, Michael A.

Skew Flat Fibrations and Totally Convex Immersions. - Ann Arbor : ProQuest Dissertations & Theses, 2017 - 45 p.

Source: Dissertation Abstracts International, Volume: 79-04(E), Section: B.

Thesis (Ph.D.)--The Pennsylvania State University, 2017.

A fibration of Rn by oriented copies of R p is called skew if no two fibers intersect nor contain parallel directions. Conditions on p and n for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of R3 by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of S3 by Gluck and Warner. We show that Salvai's classification has a topological variation which generalizes to characterize all continuous fibrations of R n by skew oriented copies of R p. We show that the space of fibrations of R 3 by skew oriented lines deformation retracts to the subspace of Hopf fibrations, and therefore has the homotopy type of a pair of disjoint copies of S2. We discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of Cn (resp. Hn) by skew oriented copies of Cp (resp. H p).

ISBN: 9780355330540Subjects--Topical Terms:

515831
Mathematics.

Skew Flat Fibrations and Totally Convex Immersions.
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100 1 $a Harrison, Michael A. $3 692215
245 1 0 $a Skew Flat Fibrations and Totally Convex Immersions.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2017
300 $a 45 p.
500 $a Source: Dissertation Abstracts International, Volume: 79-04(E), Section: B.
500 $a Adviser: Sergei Tabachnikov.
502 $a Thesis (Ph.D.)--The Pennsylvania State University, 2017.
520 $a A fibration of Rn by oriented copies of R p is called skew if no two fibers intersect nor contain parallel directions. Conditions on p and n for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of R3 by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of S3 by Gluck and Warner. We show that Salvai's classification has a topological variation which generalizes to characterize all continuous fibrations of R n by skew oriented copies of R p. We show that the space of fibrations of R 3 by skew oriented lines deformation retracts to the subspace of Hopf fibrations, and therefore has the homotopy type of a pair of disjoint copies of S2. We discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of Cn (resp. Hn) by skew oriented copies of Cp (resp. H p).
520 $a An immersion f : M → R n is called totally convex if for each pair of distinct points x, y of M, the tangent spaces at f(x) and f(y) do not contain parallel directions. We ask: given a smooth manifold M, what is the minimum dimension n = n(M) such that there exists a totally convex immersion M → R n? The corresponding question has been classically studied for immersions, embeddings, and many other maps satisfying distinguished differential conditions. The problem for totally convex immersions is especially difficult; for example, we do not know n(S 2). Wend that totally convex immersions are related to the generalized vector field problem, immersions and embeddings of real projective spaces, and the existence of nonsingular symmetric bilinear maps. We provide both upper and lower bounds on the number n..
520 $a The first chapter, dedicated to skew fibrations, is almost entirely the original work of the author and appeared in similar form in Feb. 2016 in Math. Z. The second chapter, dedicated to totally convex immersions, is similar in spirit to the treatment of totally skew embeddings by Ghomi and Tabachnikov, but the statements are technically new. The results of Chapter 2, along with the future work outlined in the introduction of Chapter 2, will be the subject of a forthcoming article.
590 $a School code: 0176.
650 4 $a Mathematics. $3 515831
690 $a 0405
710 2 $a The Pennsylvania State University. $b Mathematics. $3 2094790
773 0 $t Dissertation Abstracts International $g 79-04B(E).
790 $a 0176
791 $a Ph.D.
792 $a 2017
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10666512