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It was made clear to me in the post What answer does one get from integrating a Riemannian metric on a sphere for a great circle – does the metric’s non-flatness affect the answer? that a great circle on S2 and a plane circle (formed by a planar intersection at the equator) have the same circumference, and it doesn’t matter if a Euclidean metric or the derived Riemannian metric is used to calculate circumference of a great circle on a sphere. The metrics are the same algebraically and their scalar values are the same. The Riemannian metric is non-flat by virtue of it being on a curved space, S2.

I then found out manifolds such as exotic spheres have unique properties, some involving curvature and the metric. It occurred to me that a plane circle resulting from an intersection of such a sphere would not have the same circumference as that same circle when it resides on the sphere. I do not know what a planar intersection of an exotic sphere looks like, or what the analog of great circle is on such a sphere. I can therefore not specify this question with mathematical accuracy. However, the general idea can be applied to any manifold of dimension four or higher, not just exotic spheres, for arc lengths of curves.

For higher dimensional manifolds do the Riemannian metric and analogous Euclidean metric (from which the Riemannian metric is derived) have different scalar values? Is the Riemannian metric no longer derivable from the Euclidean one for manifolds of dimension 4 or higher, or in special dimensions such as 7D?

GaryW
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    Not sure what you mean by scalar values of a metric, but (i) Any two smooth simple curves in Riemannian geometry are locally isometric via arclength parametrizations; (ii) Two smooth simple closed curves in Riemannian geometry are isometric if and only if they have the same arc length. The ambient manifolds/metrics are immaterial. – Andrew D. Hwang May 02 '22 at 21:36
  • I meant that geimetrically a Euclidean metric is straight while a Riemannian one is curved, if it exists in a curved space. Non-zero riemann curvature tensor. In the example of great circle and plain circle, circumference is 2 pi r. Same answer after integrating either metric-the scalar values are equal. – GaryW May 03 '22 at 02:59
  • Riemanns discovery, revealing many geometries, and confirming the intrinsic nature of curvature in higher dimensions, might (so i thought) have led to discovery of some higher dimensional curves whose arc length was different when based on a non-flat metric. The geometry of metric changed its scalar value (eg circumference <> 2 pi r). At least it must have provided a single integral in place of tedious calculations if the parameterization had to be one that had a flat euclidean metric. – GaryW May 03 '22 at 03:06
  • I understand that isometries exist for a given parameterization. I thought that exotic spheres might be exceptions bc i read the notion of curvature has added significance for them. I suppose not. But riemanns work opened up many new fields so i thought non-flat metrics might have more significance than "many geometries are possible," the usual label aoplied to his famous habilitation lecture. Sorry to perseverate, especially bc that is opunion based. – GaryW May 03 '22 at 03:17
  • Im referring to the derivation explanation in spivaks intro to Diff. Geometry. The components of the metric tensor expressed by quadratic forms. Those then expressed as inner products of tangent vectors. That certainly made arc length easier to calculate for complex curves in hyperspace. Then again people before riemann hadnt really considered those higher dimensions and curvature at all. – GaryW May 03 '22 at 03:39
  • Just to note: Riemannian curvature is a two-dimensional phenomenon due to non-commuting of covariant derivatives. A real curve is therefore intrinsically flat because there do not exist two linearly independent tangent directions at a point. (Extrinsic curvature can be non-zero even for a real curve, but that "cannot be measured within the curve.") – Andrew D. Hwang May 03 '22 at 11:22
  • Ok, thanks. I will look into that non-commutability. A real curve is flat at any point but the surface its on may not be (if dim >=3)? If i want clarifivation on the significance of riemanns discovery of non-flat metrics, i can post a question on the History of Mathematics site. – GaryW May 03 '22 at 14:00
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    @GaryW Any curve, living in any Riemannian manifold of any dimension (and as curved as you want), is flat (for the reason mentioned by A. D. Hwang). I don't really get what "the surface its on" means, but I guess you are referring to the ambient Riemannian manifold. Also, exotic spheres are objects that are purely differential, not Riemannian, and there is no "the metric" on such spheres in the sense that there is no preferred one (while geometers agree that there is one preferred metric on the Euclidean sphere, namely the one induced by the Euclidean space they are canonically embedded in). – Didier May 04 '22 at 22:17
  • Is Riemannian curvature undefined on a real curve because only one independent tangent direction exists (non-commutabliity can't be calculated)? Or is it just the identity matrix (Kronecker delta) because the non-commutability (tangent with itself) calculation results in zero? – GaryW May 04 '22 at 23:22
  • Any curve is flat at a point, but can be non-flat over an arbitrarily small neighborhood? – GaryW May 05 '22 at 02:04
  • It is perplexing that curves are flat while surfaces may not be. The metric (sphere for example) is integrated to find arc length and it is close to, but not exactly, flat. Now i'm wondering what it means, in laymans terms, to say a manifold has a non-flat metric, be it a preferred one or not. What is the major outcome-for arc length, or understanding the nature of points on the surface? If a surface has to bend in 2 directions to have an intrinsic curved metric, does this simply not impact arc length calculation? – GaryW May 06 '22 at 16:21
  • You can close the question. I made a new post on the HIstory of Science and Mathematics stack exchange site. – GaryW May 15 '22 at 02:22

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