Talk:Complete manifold

Inadequate definition

this article defines a geodesically complete riemannian manifold (M,g) to be one that has the property (call it P) that any two points in the manifold M can be connected by a length-minimizing geodesic. i'm not sure this is the most precise definition of geodesic completeness: consider a proper open subset (say the unit open ball) in $\mathbb {R} ^{n}$ with standard euclidean metric g. then surely any two points in the ball can be connected by a length-minimizing geodesic (namely a straight line between the two points), but we don't consider the unit open ball to be geodesically complete because the spray of geodesics emanating from any point p in the ball runs out of the ball in finite time.

if we define geodesic completeness to mean that all geodesics have domain $\mathbb {R}$, rather than just an open subinterval of $\mathbb {R}$, then of course we get property P. but the example i show above seems to suggest the converse is not true. we want the more primitive notion of geodesic completeness; since geodesics can be understood as orbits of a hamiltonian vector field on the tangent bundle $TM$, this deeper definition gives a global hamiltonian flow on $TM$ and thus a smooth action of (all of) $\mathbb {R}$ on $TM$. merely assuming property P will not give this smooth action.

Mlord 21:41, 17 May 2007 (UTC)

I guess my problem is related, but I don't know: is the geodesic unique, or does it suffice that there is a set of equally long geodesics? ... said: Rursus (bork²) 09:43, 3 February 2009 (UTC)

As of this time, it appears that the article has been fixed, and whatever definition that was, was replaced by one that requires infinite length. So I guess this conversation is closed. 67.198.37.16 (talk) 03:15, 3 November 2020 (UTC)

Baffling

This article is jargon-filled, and completely opaque to a non-mathematician. I wouldn't have a clue how to fix it. Eaglizard (talk) 12:01, 20 January 2013 (UTC)

? I'm not sure that there is anything to be fixed. Unlike some topics, you sort-of have-to-have some understanding of what a geodesic is, and what a manifold is; these are bare-bones pre-requisites. If you know what a manifold is, and what a geodesic is, then all the "jargon" just melts away; its just plain-old ordinary language. I'll try to think about how to "simplify" this, but I am not sure it is possible. 67.198.37.16 (talk) 03:09, 3 November 2020 (UTC)

Vandalism!

In 2011, there was some vandalism that deleted some basic definitions. The vandalism was uncaught at the time. The content should be restored. I'd do it, but I'm busy with other things. (The deleted content is not really all that important; it just states a few basic definitions.) 67.198.37.16 (talk) 03:23, 3 November 2020 (UTC)

Requested move 27 June 2024

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: moved. (closed by non-admin page mover) BilledMammal (talk) 18:44, 4 July 2024 (UTC)

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Geodesic manifold → Complete manifold – "Geodesic manifold" is an extremely unusual term, and this page doesn't even use it because it is so unusual. The correct term would be either "Complete manifold" or "Geodesically complete manifold". Mathwriter2718 (talk) 18:09, 27 June 2024 (UTC)

Support I've seen this term only for a submanifold where every geodesic is a geodesic in the ambient manifold, such as in [1]. –LaundryPizza03 (dc̄) 16:41, 28 June 2024 (UTC)

The article was completely rewritten in 2011, but the old content seems to be mathematically indistinct from the new content. –LaundryPizza03 (dc̄) 16:44, 28 June 2024 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.