|WikiProject Mathematics||(Rated B+ class, High-importance)|
Changed Euclidean space to linear space
I edited the article to remove references to Euclidean space (to which the manifold is locally similar) which might be read to imply there is a local orthogonality property or equivalently a metric structure. I added a sentence in the first paragraph to clarify this important distinction. Regards, James Baugh (talk) 22:28, 5 January 2011 (UTC)
The lede currently starts "A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas." This really does not tell us much about differentiable manifolds, particularly for someone who does not know yet what a manifold is. The familiar refrain about "collection of charts" is particularly unhelpful. Who can honestly claim that he or she thinks of a manifold as a "collection of charts"? Rather, the best definition is the one found, for example, in Arnold. This says that a finite-dimensional differentiable manifold is a subset of Euclidean space of a suitable (larger) dimension. The condition is that it is locally the graph of a (possibly vector-valued) differentiable function. Tkuvho (talk) 08:50, 31 May 2012 (UTC)
Someone who apparently thinks there's a convention against italics in subscripts and superscripts did a lot of work here. Please see WP:MOSMATH. Variables are italicized; digits, parentheses, and things like det, log, max, cos, etc., are not. Michael Hardy (talk) 21:34, 9 July 2012 (UTC)
Re: Tensor bundle not being a differentiable manifold
Removed the following statement under the subsection Tensor Bundle:
"The tensor bundle cannot be a differentiable manifold, since it is infinite dimensional."
It's clearly incorrect as stated. I couldn't find any source or support to the claim that the tensor bundle is in general not a diff. manifold. In addition, the reasoning contradicts the later mention of Banach and Hilbert manifolds as infinite-dimensional diff. manifolds (which are both well-established constructions).
- I think it suffices to say that the tensor bundle is not a differentiable manifold in the traditional sense, because it is infinite dimensional. (It's also not a Banach or Hilbert manifold, by the Baire category theorem.) Sławomir
Biały 11:51, 4 February 2016 (UTC)
The comment(s) below were originally left at several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section., and are posted here for posterity. Following
|Recommend moving most of this article to new smooth manifold page. Geometry guy 23:14, 14 April 2007 (UTC)|
Last edited at 23:14, 14 April 2007 (UTC). Substituted at 01:59, 5 May 2016 (UTC)