# Talk:Metric tensor

WikiProject Mathematics (Rated C-class, High-priority)

## Request

Please do not delete this page. although there is an alternative approach to differential geometry, the component-based approach is fundamental to understanding the 'modern' approach, and the metric tensor is the fundamental definition of Riemannian geometry. What are the goals of this encyclopedia? what should they be? to be esoteric and create what some very few people might find to be 'elegant' and 'precise', or to make information accessible? I believe that it is the latter. Furthermore, I do not believe that the two goals are mutually exclusive. I believe, rather, that writting in a clear language that can readily be understood is a form of eloquence and 'perfection', and should be a priority. I am reminded of early medicine, when the professors turned the science of medicine into an esoteric and pedantic rite in pursuit of the luster of exclusive power. I would hate to see mathematics go the same way.

I like your attitude. I don't suppose you know what a tensor product is? See my comment on Talk:Tensor product. By the way, Kevin, you should sign your entries on talk with ~~~~, which is automatically replaced with a signature like the following. -- Tim Starling 04:23 Mar 14, 2003 (UTC)

perhaps we should explain the implicit summation and products of differentials more? - Gauge 05:41, 31 Jul 2004 (UTC)

This page is simple, clear, and essentially self-contained. Browsing from the General Relativity entry, I was much happier with this page than with most other tensor-related explanations, which were so extravagantly reference-dependent as to be useless. I have a pretty solid general math and physics background, and doubt that a much more demanding presentation would serve a significant number of readers. Peter 19:50, 4 Feb 2005 (UTC)

I wonder if anybody has thought about the divide and conquer approach for writing mathematical objects like equations. It requires only the existence of parallel computers normally used in business (say stores).

Benjamin Cuong P. Nghiem bcnghiem@hotmail.com

This article is true to the intuition of Gauss and the other pioneers of the subject - it deals with the topic as they dealt with it at the time. This makes it more readable and understandable, more suitable for learning. Such intuition is lost almost entirely from the more formal, abstract style that characterises most WP articles in this area. Such entries are less suitable for learning, and I suspect make sense only to people who already understand the subject. Whilst I see the intellectual appeal of recasting historical fundamental works in a modern abstract form, reducing it to its elements, something really important is lost in that process which may not even occur to the mechanical abstract mind, namely the human intuition and genius of the original pioneers (here Gauss). I wish all the articles on differential geometry, general relativity and perhaps wider could somehow retain the intuition, genius and creativity of the pioneers, to help learning and to inspire others. Therefore I hope this article stays! Phrichuk (talk) 12:33, 24 January 2022

## Clarification possible?

The definition says

More precisely, given any open subset U of manifold M and any (smooth) vector fields X and Y on U ...

Then, as far as I understand the next two lines, elements of X and Y end up as arguments of g_p, though g_p is only defined on the tangent space at p. Should this not go into the definition of the vector fields X and Y. And why are they defined "on U", which is exactly not the tangent space of a p\in U? It would be great if this could be clarified a bit.

As you can guess, I am not an expert, just someone trying to understand this :-) Haraldkir (talk) 13:24, 17 August 2019 (UTC)

## possible simplafications for the opening pharagraph

I got super confused while reading the first paragraph, the wording doesn't make sense, "as input a pair of tangent vectors" shouldn't it be "an input of a pair of tangent vectors." and if not, than why "as" and how does it apply to metric tensors. Changing As to An makes the page flow more correspondingly. Also this line should be removed, "(or higher dimensional differentiable manifold)" it doesn't really help make the paragraph make sense. While yes Metric Tensors do apply do higher dimensional differential manifolds it shouldn't really be added to an introduction paragraph in brackets. It makes the page look blocky and messy. It should either be added to the other elaborating paragraphs after the introduction or have its own sentence explaining it instead of it being in brackets. My solution is a changing of the opening, something like this, "In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes an input of a pair of tangent vectors v and w at a point of a surface and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. Metric tensors also apply to 3rd Dimension differentiable manifold as well as higher dimension differential manifolds that exist in 4D spaces. In the same way as a dot product, metric tensors are used to define the length of tangent vectors and angle between them. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold." Though my sentence uses a lot of words starting with D, making saying it or even reading it a bit of a tongue twister it makes sense, and could be simplified more with some commas.

In conclusion a simplification of the opening paragraph is very much possible, and my possible changes could make it a more friendlier page to read, as it really relies on people already knowing what a metric tensor is. JustAnEagerLearner (talk) 21:27, 24 April 2022 (UTC)

I agree that the first paragraph was pedantic. However, it was correct, as "taking as input" is a common jargon when talking of functions. Your edit is not an improvement, as "one definition of a metric tensor is a type of function" is non-sensical (a function is not a definition).
I have rewritten the first paragraph, for making it easier to understand for non-specialists. D.Lazard (talk) 10:38, 25 April 2022 (UTC)
Simplifying is good, but only if it preserves accuracy. Also, wikilinks such as  [[Vectors (mathematics and physics)]] should generally be piped to avoid clutter, e.g., [[Vectors (mathematics and physics)|]]. 4D is just a special case of higher dimension, albeit an important one, and probably does not belong in the lead. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:05, 25 April 2022 (UTC)

## Distance

There are fundamental problems with the text On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d(p, q) whose value at a pair of points p and q is the distance from p to q.:

1. It is circular; presumably the author intended to say the length of a geodesic from p to q.
2. There may be no geodesic from p to q.
3. There may be multiple geodesics from p to q, of differing length.

I propose the text On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic. If there is a geodesic between two points p and q whose length is globally a minimum, then the distance d(p, q) between the points is defined to be that length; otherwise d(p, q) is defined to be the greatest lower bound of all the smooth curves connecting them. Equipped with this notion of length, a Riemannian manifold is a metric space with the distance function d(p, q), i.e., d satisfies the identity, symmetry and triangle inequality axioms. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:42, 25 April 2022 (UTC)

Notes