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I am trying to understand the following to defintions of being "intrinsic". They are from Differential Geometry of Curves and Surfaces by Tapp.

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Question:

  • In what way are these defintions consistent?
  • To call quantity intrinsic if it can be expressed completely in terms of the first fundamental form makes sense.
  • To me it seems that requiering invariance w.r.t. isometries captures different properties that I would expect from intrinisc properties. For example that deforming a surfaces should not alter the length of a curve on that surfaces.

It seems to me that there are two types of intrinisc properties and that the "intrinisic geometry" consists out of these.

Many thanks in advance for any hints or explanations!

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Hans
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The definitions are consistent in the sense that a measure that is completely described in terms of the first fundamental form is preserved by isometries, since isometries preserve the first fundamental form, so the second definition "contains" the first one.

You might ask yourself what happens with the measurements that are preserved by isometries but are independent of the first fundamental form:

  1. You could say that those measurements depend on the first fundamental form on a "trivial way", but this might be a bit dangerous.

  2. An isometry is in particular a diffeomorphism, so the measurements that it preserves are precisely those that are described by the first fundamental form, or those that are preserved by diffeomorphisms, that is, the topological and differential structures of the space.

So the way I suggest you think about it is the following: an isometry is a map that preserves

  • The set structure (number of points): it is a bijection

  • The topology: it is a homeomorphism

  • The differentiable structure (which functions on the manifold are differentiable): it is a diffeomorphism

  • The measurements or geometrical properties (the first fundamental form): it is an isometry

Then the first definition is focusing on this last point, and the second contemplates all of them. Notice that each condition is stronger than the previous one and contains it.

This can be generalised to any dimensions by swapping regular surfaces for Riemannian manifolds and first fundamental form for Riemannian metric.

From your last point, the word "deforming" ought to be made precise, but as long as the deformation is a bijection, a homeomorphism or a diffeomorphism (or an isometry, of course), it is contained in the definition of isometry.

Let me know in the comments whether this answers the question :)

ashsan98
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    Thank you for your answer. What you wrote in the first paragraph definetly makes sense. The second part doesn't really make sense to me. What I don't understand is why is there a map introduced. One way I can make sense of it is by saying it is supposed to be analogous to Euclidean geometry, i.e. the study of "things" that are invariant under rigid motions. In other words a definition of a "new" geometry, namely the "intrinisc" one. I also found a definition of intrinisic geometry in terms of charts and $g_{ij}$ on $\mathcal{R}^{n}$, which at least involves a map too. – Hans Jun 09 '23 at 09:58
  • @Hans that is actually an important point: it has to do with a way of thinking in mathematics that was popularised by Klein under the name Erlangen Program (read about it!). The key idea is that an easier way to study geometric properties of a space is to look at the group of symmetries that preserve those properties. For example, you can say that the topology of a space is the property that is preserved by topological mappings, i.e. continuous functions, and define these functions instead of defining what a topology is. Similar approaches permeate maths (like Yoneda in category theory). – ashsan98 Jun 09 '23 at 10:40
  • @Hans for example, as you correctly said, Euclidean geometry refers to properties or constructions that are preserved by the Euclidean group, i.e. the group of rigid motions. So you can explicitly define and study rigid motions instead of defining and studying Euclidean invariants, and vice versa. Klein suggested the study of mappings because you normally have more structure associated to them (i.e. group structure), and that might make them easier to work with. – ashsan98 Jun 09 '23 at 10:43
  • Okay, then I would say that being intrinisic,i.e. depending only on $g_{ij}$ and being invaraint under isometries,i.e. being "in" the/part of the intrinsic geometry (can you say that?) can be seen a priori as two seperate things. But as you mentioned, since depending only on $g_{ij}$ implies invariance, these notions are in the/part of the intrinsic geometry. I added a pricture to my question. How does this definition fit in? – Hans Jun 09 '23 at 11:25
  • @Hans the notation does not neccesarily allow you to say being in the intrinsic geometry; you rather say being an intrinsic geometric property or being intrinsic: there is no confusion with the other notion because of their equivalence a posteriori. In your new picture, g is the first fundamental form and you basically say that things described by your fundamental form are independent of your choice of parametrisation, because a choice of one changes the first fundamental form and the description of the properties themselves in terms of the coordinates accordingly – ashsan98 Jun 09 '23 at 11:47
  • @Hans once you choose a parametrisation, passing to another one and modifying the metric g accordingly corresponds to performing an isometry – ashsan98 Jun 09 '23 at 11:52
  • Why are they equivalent, i.e. how does being invariant under isometries imply dependence only on $g_{ij}$? I don't see how that can be shown or do you mean by equivalent a posteriori, that if we define intrinsic $:\iff$ invariant under isometries, then depending on $g_{ij}$ $\iff$ intrinsic.But this definition leads me back to square one, meaning what "intrinsic" properties are captured by that more generalised notion of "intrinsic". It seems to me a bit like a „leap of faith“ to call invariance intrinsic in the sense that considering only constructions that depend only on $g_{ij}$ – Hans Jun 09 '23 at 13:45
  • might be a bit too restrictive, but at least it includes the „intuitive $g_{ij}$“ version. So referring to your answer are the things you listed to be considered intrinsic properties? I am looking for examples that motivate the „invariance approach“ to intrinsic.

    The thing with the coordinate charts I am not sure about. If I think about it, I would say it is a good model for „staying in the the surface“.

     – Hans Jun 09 '23 at 13:46
  • Choosing a chart around point and then consider notions that depend on $g_{ij}$ only is the I would say the same thing as „looking at the manifold M from $T_{p}M$“ and consider everything in terms of the metric on M. I don’t see why you looking at transition maps that are isometries. Again, I am looking for a motivation to define intrinsic in this why and how all these different (at least to me) approaches can be unified in the definition in terms of isometries. I just can’t get my head around it. Sorry for the endless questions. :/ – Hans Jun 09 '23 at 13:46
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    @Hans I think I cannot add much more than what I said to the discussion, perhaps only formulate it in other words... I would suggest you work with some examples to develop an intuition for what is invariant and what not, and then come back to the question to see if you understand it better. Otherwise, I believe your doubts are very complicated to grasp without some particular cases in mind. I would be happy to keep answering but I believe it would not help you to keep it so abstract, but you could maybe add an example and discuss it – ashsan98 Jun 13 '23 at 10:28