From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1177 Path: news.gmane.org!not-for-mail From: Andree Ehresmann Newsgroups: gmane.science.mathematics.categories Subject: Answer to Davilov Date: Wed, 21 Jul 1999 17:58:09 +0200 Message-ID: <1.5.4.16.19990721180227.1c4f8688@mailx.sc.u-picardie.fr> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Trace: ger.gmane.org 1241017618 29827 80.91.229.2 (29 Apr 2009 15:06:58 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:06:58 +0000 (UTC) To: cat-dist@mta.ca Original-X-From: cat-dist Thu Jul 22 12:16:12 1999 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id KAA08115 for categories-list; Thu, 22 Jul 1999 10:46:49 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Received: from porky.mta.ca (porky.mta.ca [138.73.101.9]) by mailserv.mta.ca (8.9.3/8.9.3) with SMTP id MAA28330 for ; Wed, 21 Jul 1999 12:58:16 -0300 (ADT) X-Received: FROM gip.u-picardie.fr BY porky.mta.ca ; Wed Jul 21 13:02:48 1999 X-Received: from ppp1.u-picardie.fr (ppp1.u-picardie.fr [194.57.109.97]) by gip.u-picardie.fr (8.9.1/8.9.1) with SMTP id RAA07166 for ; Wed, 21 Jul 1999 17:58:09 +0200 X-Sender: ehres@mailx.sc.u-picardie.fr X-Mailer: Windows Eudora Light Version 1.5.4 (16) Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 44 Xref: news.gmane.org gmane.science.mathematics.categories:1177 Archived-At: Davilov writes: >Given an object M in the ``normal'' category of finitely dimensional >smooth manifolds Man (not in SDG sense), what it the universal property >of the tangent bundle TM? Several authors have been interested in this problem many years ago, that has led to the study of the functors from Man to the category of vector bundles. Such functors are completely characterized in Epstein, "Natural vector bundles", Lecture Notes in Math. 99, Springer 1969, p. 171-195. >>From his theorems, it results in particular that Ehresmann's first order velocity functors Tn which associate to a manifold M the vector bundle of the 1-jets from R^n to M (in particular T for n=1) are the only such product preserving functors. This result is generalized in an abstract setting to characterize connections in: Bowshell, "Abstract velocity functors", Cahiers de Topologie et Geom. Diff. XII-1 (1971), 57-82. Later on, related (or more general) problems are considered in: Palais & Terng, "Natural bundles have finte order", Topology 16 (1977), 271-277, Epstein & Thurston, "Transformation groups and natural bundles", Proc. London Math. Soc. 38 (1979), 219-236. Eck, "Product preserving functors on smooth manifolds, J. Pure & App. Algebra 42 (1986), 133-140. Kolar, An abstract characterization of the jet spaces, Cahiers de Topologie et Geom. Diff. XXXIV-2 (1993), 121-125. Dupovec & Kolar, On the jets of fibered manifold morphisms, Cahiers de Topologie et Geom. Diff. XXXIV-2 (1993), 121-125. Hoping these old references can be of some interest Best regards Andree C. Ehresmann