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Motivation (update): I am interested in properties/structures/objects that are determined by the metric alone, but are not among the usual ones that we call intrinsic, like Levi-Civita connection, Riemann curvature, or anything that is locally determined by the metric. The only kind of such properties I could find are relational, determined by the metric in a non-local way, like the following example, so I am interested if I can find something like this researched in the literature.


Let $M$ be a Riemannian manifold. Its geometric properties, those which depend on the metric, not topology etc., are usually divided in two classes: intrinsic (depending on the metric of $M$ alone) and extrinsic (depending on an embedding of $M$ into a higher-dimensional Riemannian manifold).

I am thinking that there may be some "intermediate" properties, which are geometric, but neither purely intrinsic nor purely extrinsic. As an example, consider the surface $z=\sin x\cos y$ in $\mathbb R^3$:

Graph of the function $\sin x\cos y$

Choose a side of the surface. We can see that there are regions where the surface is either convex or concave (with respect to the chosen side of the surface). Clearly here "convex" and "concave" is a property which depends on the embedding and on the side of the surface we choose as reference. These regions are represented below with alternating colors:

enter image description here

Every such region is isometric with any other, being them convex or concave.

We can define now the following relation of equivalence between two such regions $A,B\subset M$: $A\sim B$ iff in an embedding in $\mathbb R^3$, $A$ and $B$ are either both convex, or both concave.

Now this relation $\sim$ is not reducible to the metric, because $A$ and $B$ are isometric even if $A\nsim B$. So it is not intrinsic. But neither it is extrinsic, since it doesn't depend on the embedding (although the embedding made us realize the difference between the two kinds of regions).

I realize now that it is not right to say $\sim$ is not intrinsic, since it can be derived from the metric. But you can't know to what equivalence class such a region belongs based on its metric alone, since if we only have two pieces of the manifold we can't say only based on the metric locally if they are equivalent or not even if they are isometric, we need to know how these regions are connected to another.

So let us say that this kind of property is not "locally intrinsic", rather that it is not intrinsic, since it depends on the metric, but knowing the metric locally is not enough.

Now consider that we turn the surface into a cylinder (topologically speaking), by cutting a strip which contains on each row an odd number of such regions, and gluing the opposite edges (without shifting or twisting the surface). This new "cylindrical" surface is smooth, and it will make the two equivalence classes of region determined by $\sim$ become a single class. I believe this cylindrical surface can't be embedded isometrically into $\mathbb R^3$. But can it be embedded isometrically into a higher dimensional Euclidean space?

My questions are:

  1. Can you provide some references in Riemannian geometry which study this relation?
  2. Are there other geometric properties like this one, which are not locally intrinsic, nor extrinsic?

Update 2 I realized how to make the question more specific:

2'. I am interested in references to geometric properties/structures/objects which we encounter when studying Riemannian submanifolds, but are independent on the embedding (considering that the dimension of the ambient manifold is fixed).

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    $\begingroup$ If I understand correctly, you are interested in notions that are not intrinsic to a raw Riemannian manifold, but become intrinsic if one also attaches a second fundamental form to that manifold as an additional structure? The pairing of a Riemannian manifold with a second fundamental form shows up in general relativity as the natural "initial data" for evolution equations such as the Einstein equations, see e.g. arxiv.org/abs/1304.1960 . One subtlety in that context is the presence of nontrivial constraint equations relating the form with the metric. $\endgroup$ – Terry Tao Aug 30 '17 at 17:58
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    $\begingroup$ This is tangential, but may I ask: What leads you to "believe this cylindrical surface can't be embedded isometrically into $\mathbb{R}^3$? I don't doubt you; I would just like to understand the reasoning. $\endgroup$ – Joseph O'Rourke Aug 30 '17 at 21:51
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    $\begingroup$ I don't know if this is quite what you are looking for, but the induced metric on a non-flat minimal surface in $R^3$ satisfies (at points of non-zero Gauss curvature -- i.e. most of them) the so called Ricci condition: $\Delta_g \log |K_g| =4 K_g$ -- this is a purely intrinsic equation. Remarkably, any metric on a simply connected 2D domain satisfying $K_g<0$ and the Ricci condition gives you a minimal immersion into $R^3$. $\endgroup$ – Rbega Aug 30 '17 at 23:02
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    $\begingroup$ I should add that the induced metric does not determine the minimal immersion uniquely. Even after modding out by natural symmetries of $R^3$ there is still an $S^1$ family of different immersions -- the so-called associate family. However, if you know the second fundamental form, this pins down the immersion (mod symmetries of $R^3$). $\endgroup$ – Rbega Aug 30 '17 at 23:07
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    $\begingroup$ I started reading this question before I looked at the author, and I was surprised to learn that it wasn't posted by @JosephO'Rourke. (This is intended as a favorable comparison.) Interesting stuff! $\endgroup$ – Paul Siegel Aug 30 '17 at 23:15

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