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I'm not very happy with any

This question is still wide open - all of the answers so far rely on magical calculations. I've only accepted an answer because, and I'm about to lose control over by bounty rules, otherwise one would be accepted answersautomatically. I'm thinking of accepting I can't change the Andrey's accepted answeras the most promising conceptually, but unfortunately I don't have the time to think about it very carefully anymore -- I've taught the class already. Feel free would be amazing to comment have more discussion on itthis question.

I'd like a nice proof (or a convincing demonstration), for a surface in $\mathbb R^3$, that explains why the following notions are equivalent:

1) Curvature, as defined by the area of the sphere that Gauss map traces out on a region.

1.5) The integral of the product of principal curvatures.

2) The angle defect of parallel transport about a geodesic triangle.

(This equivalence may be considered as either a part of the Theorema Egregium or a part of Gauss-Bonnet. Proving that numbers 1 and 1.5 are the same is pretty easy).

Motivation: I'm teaching a five-day class for very bright high-school students. The idea is to give them an impression of what geometry is about. However, when I looked at Spivak's proof of this, it was much more of a messy calculation than I expected. I'd like, if at all possible, something more conceptual, ideally with a nice picture attached to it.

Since this doesn't have to be a perfectly complete class, I'll be perfectly happy with a good illustration of why this is true instead of a rigorous proof, if a conceptual and rigorous proof is completely out of the question.

One idea I had is to show the example of a sphere and the hyperbolic plane, and then explain that on very small scales the curvature is constant. However, then I would need a nice proof that the embeddings of the hyperbolic plane in $\mathbb R^3$ have curvature -1.

Thank you very much!

P.S. This question is related, but not quite the same (I hope), to this question: http://mathoverflow.net/questions/22410/equivalent-definitions-of-gaussian-curvature

P.P.S. Thank you to whomever recommended to Berger's "Panoramic View of Riemannian Geometry". it was quite useful to me. I do not know why you deleted your answer.

That books claims there is no conceptual proof. However, I'd still be very happy with a nice illustration of why one should believe this, especially for negative curvature.
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[STILL OPEN] Why are two notions of Gaussian curvature are the same - what is the simplest & most didactic proof?

I'm not very happy with any of the answers, and I'm about to lose control over accepted answers. I'm thinking of accepting the Andrey's answer as the most promising conceptually, but unfortunately I don't have the time to think about it very carefully anymore -- I've taught the class already. Feel free to comment on it.

I'd like a nice proof (or a convincing demonstration), for a surface in $\mathbb R^3$, that explains why the following notions are equivalent:

1) Curvature, as defined by the area of the sphere that Gauss map traces out on a region.

1.5) The integral of the product of principal curvatures.

2) The angle defect of parallel transport about a geodesic triangle.

(This equivalence may be considered as either a part of the Theorema Egregium or a part of Gauss-Bonnet. Proving that numbers 1 and 1.5 are the same is pretty easy).

Motivation: I'm teaching a five-day class for very bright high-school students. The idea is to give them an impression of what geometry is about. However, when I looked at Spivak's proof of this, it was much more of a messy calculation than I expected. I'd like, if at all possible, something more conceptual, ideally with a nice picture attached to it.

Since this doesn't have to be a perfectly complete class, I'll be perfectly happy with a good illustration of why this is true instead of a rigorous proof, if a conceptual and rigorous proof is completely out of the question.

One idea I had is to show the example of a sphere and the hyperbolic plane, and then explain that on very small scales the curvature is constant. However, then I would need a nice proof that the embeddings of the hyperbolic plane in $\mathbb R^3$ have curvature -1.

Thank you very much!

P.S. This question is related, but not quite the same (I hope), to this question: http://mathoverflow.net/questions/22410/equivalent-definitions-of-gaussian-curvature

P.P.S. Thank you to whomever recommended to Berger's "Panoramic View of Riemannian Geometry". it was quite useful to me. I do not know why you deleted your answer.

That books claims there is no conceptual proof. However, I'd still be very happy with a nice illustration of why one should believe this, especially for negative curvature.
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I'd like a nice proof (or a convincing demonstration), for a surface in $\mathbb R^3$, that explains why the following notions are equivalent:

1) Curvature, as defined by the area of the sphere that Gauss map traces out on a region.

1.5) The integral of the product of principal curvatures.

2) The angle defect of parallel transport about a geodesic triangle.

(This equivalence may be considered as either a part of the Theorema Egregium or a part of Gauss-Bonnet. Proving that numbers 1 and 1.5 are the same is pretty easy).

Motivation: I'm teaching a five-day class for very bright high-school students. The idea is to give them an impression of what geometry is about. However, when I looked at Spivak's proof of this, it was much more of a messy calculation than I expected. I'd like, if at all possible, something more conceptual, ideally with a nice picture attached to it.

Since this doesn't have to be a perfectly complete class, I'll be perfectly happy with a good illustration of why this is true instead of a rigorous proof, if a conceptual and rigorous proof is completely out of the question.

One idea I had is to show the example of a sphere and the hyperbolic plane, and then explain that on very small scales the curvature is constant. However, then I would need a nice proof that the embeddings of the hyperbolic plane in $\mathbb R^3$ have curvature -1.

Thank you very much!

P.S. This question is related, but not quite the same (I hope), to this question: http://mathoverflow.net/questions/22410/equivalent-definitions-of-gaussian-curvature

P.P.S. Thank you to whomever recommended to Berger's "Panoramic View of Riemannian Geometry". it was quite useful to me. I do not know why you deleted your answer.

That books claims there is no conceptual proof. However, I'd still be very happy with a nice illustration of why one should believe this, especially for negative curvature.
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