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I wondering if there is a model of signed curvature in n-dimensions for a tangent n-ball, the reason being that I would like to investigation a representation of manifolds by reconstructing any closed shape at least in Euclidean space using only information about its curvature and initial conditions.

Is there a theory I can rely on for this or do I not even need signed curvature?

OneWhoBlends
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    I do not know what the expression "model of signed curvature" means, or even the expression "signed curvature". Could you give a reference to a publication, blog or wiki using these terms where they are explained in more detail. – Ben McKay Aug 27 '20 at 18:35
  • I don't understand why signed curvature is confusing to specifically mathematicians (and no one else). A vector relative to a closed surface can point either inward or outward, this has been the basis of many advancements in computer graphics and physics for decades. Similarly, a vector drawn from the tangent point to the radius of a tangent n-ball may point inward or outward. https://en.wikipedia.org/wiki/Curvature mentions signed curvature. – OneWhoBlends Aug 27 '20 at 18:49
  • The mean curvature is sign dependent but it is not an intrinsic invariant. Globally, reconstruction may not be possible. The flat torus and the Euclidean spaces are examples of spaces with identical trivial curvatures – Liviu Nicolaescu Aug 27 '20 at 19:23
  • Alright, well is there a set of minimal supplementary conditions that can be added which would guarantee reconstruction? – OneWhoBlends Aug 27 '20 at 19:25
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    What's not clear here is how much differential geometry you've already studied. Could you clarify that? It'll be easier to guide you if we get a sense of that. – Deane Yang Aug 27 '20 at 21:40
  • I don't see how that is relevant to a yes/no question. If you have a good explanation for that, I can provide additional information, otherwise I consider it off topic. – OneWhoBlends Aug 27 '20 at 21:46
  • The expression "signed curvature" remains ambiguous because the reference you gave only defines signed curvature for curves in the plane. You are using the term "signed curvature" in reference to manifolds of apparently arbitrary dimension in Euclidean space, and perhaps to more general Riemannian manifolds, but that is not clear in your question. – Ben McKay Sep 27 '20 at 07:12
  • What is this tangent $n$-ball? Do you mean that you suppose that you know of the existence of a Riemannian manifold, and you know the sectional curvature of all of its metric balls tangent to any given hyperplane, as a function of radius of ball, and you want to know what Riemannian manifold it is, up to isometry, or up to diffeomorphism, or up to homotopy equivalence? – Ben McKay Sep 27 '20 at 07:17
  • I don't see how there is ambiguity in a tangent $n$-ball. In a 2D plane you have a tangent circle, in 3D you have a tangent sphere. From there you can see the pattern. – OneWhoBlends Sep 27 '20 at 18:57
  • in the wikipedia article you cited above is applied to a curve only and not to a higher dimensional geometric space. Looking at what you've written in both your post and comments, it looks to me like what you want is indeed the second fundamental form. This assigns to each unit tangent vector (i.e., an element of the tangent $n$-sphere), a number that could be called the signed curvature in that direction. – Deane Yang Sep 27 '20 at 22:14
  • As for reconstructing the surface, there is a theorem that says roughly that if you know the first and second fundamental forms of a surface, the position of one point of the surface in Euclidean space, and the position of the tangent plane at that point, then there is a unique embedding of the surface into Euclidean space with the prescribed first and second fundamental form. – Deane Yang Sep 27 '20 at 22:21
  • Here is a nice explanation of this theorem by Robert Bryant. https://mathoverflow.net/a/155755/613 – Deane Yang Sep 27 '20 at 22:24
  • In the article they definitely include a vector-valued model, that is what the bolded letters are for. – OneWhoBlends Sep 28 '20 at 03:43

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I think you are probably looking to find an analogue of the Serret--Frenet theory for higher dimensional submanifolds of Euclidean space; this is discussed in detail in What is the analog of the "Fundamental Theorem of Space Curves," for surfaces, and beyond?

The essential difficulty in making this question precise is that the curvature has to be somehow "given". But, if we take curvature to mean the Riemann curvature tensor, for example, or one of its components (or the sectional curvature), then it is not clear how, without knowing the manifold $M$, we can explicitly describe a tensor living on that manifold (or living on a Grassmann bundle over that manifold). To write out the curvature tensor, you need to parameterize the manifold, and then you will at least be assuming that you know the diffeomorphism type of the manifold, which seems to be something you want to assume unknown. One approach to avoid knowing the diffeomorphism type of the manifold: assume quasihomogeneity of the curvature tensor (or the Ricci tensor, etc.), i.e. that any two points of the manifold have the same curvature tensor (or Ricci tensor, etc.) under some linear isometry of tangent spaces. But this hypothesis could be too strong; just consider surfaces. Another approach assumes the diffeomorphism type is known, for example the problem of prescribed scalar or Ricci curvature on a given manifold, for which there are results. But I don't think there are results for prescribing the sectional curvature or the entire Riemann curvature tensor. To make your question clearer, please give some details about what you want to assume is known, and how the known information is presented.

Ben McKay
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  • Thank you for the link, that looks interesting. So in this sense, a "Frenet frame" is a broader initial condition needed to reconstruct a manifold from only curvature information? – OneWhoBlends Aug 27 '20 at 18:51
  • @OneWhoBlends, "Frenet frame" is not a condition. It's something you can define on a surface or higher dimensional submanifold in Euclidean space. – Deane Yang Aug 27 '20 at 21:39
  • Right, it's obviously broader than the scope of just an initial condition, but much like an initial condition, it is an additional piece of information that precisely resolves the otherwise ambiguous possibilities of a differential solution. The only question remaining is: does that additional information allow one to reconstruct a surface? – OneWhoBlends Aug 27 '20 at 21:41
  • If you have an abstract manifold with Riemannian metric and a symmetric tensor that satisfies the Gauss and Codazzi-Mainardi equations, then that determines an isometric embedding (I.e., a reconstruction), unique up to a rigid motion of the manifold into Euclidean space as a hypersurface, such that the symmetric tensor is the second fundamental form. Is that what you’re looking for? – Deane Yang Aug 28 '20 at 00:38
  • It sounds like what you call signed curvature on the unit sphere in the tangent space is the second fundamental form. Is that right? – Deane Yang Aug 28 '20 at 00:39