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Which of principal curvature, Gaussian curvature and mean curvature is intrinsic and why? Which book will have this explanation?

I read a book (Remannian Geometry, Do Carmo, p129). It has a saying: "The symmetric functions of $\lambda_1$,..., $\lambda_2$ are invariants of the immersion." What did he mean? Does he means that the principal curvature is the intrinsic property?

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Why not simply read the Wikipedia entry on curvature? en.wikipedia.org/wiki/Curvature –  heropup May 14 at 9:54
    
@heropup I know only Gaussian curvature is intrinsic, but I also have a qustion about the saying in Do Carmo's textbook. What did he mean? –  user34669 May 14 at 10:42

1 Answer 1

Well, this question has been already answered here but I will try to rephrase it to be more specific.

The principal curvatures are the eigenvalues of the shape operator which is essentially a derivative of the Gauss map, and the Gauss map in its turn is essentially the unit normal (I speak about surfaces in three dimensional space, for certainty). You know the unit normal when you know the embedding (and the orientations). In other words, you need the extrinsic information (=immersion, embedding) to proceed.

This means that by their nature the principal curvatures are extrinsic invariants, and everything that we are able to make out of them should be expected to be extrinsic, that is dependent on the embedding (well, locally an immersion is an embedding). "Invariants of the immersion" is a synonym for "extrinsic invariants".

The word "intrinsic" means that you only need to know the first fundamental form in order to compute such an intrinsic quantity. Sometimes the first fundamental form is called the intrinsic metric of the surface (everywhere by a metric I understand a Riemannian metric).

The Gaussian curvature is an exceptional quantity which, on the one hand, can be expressed as a symmetric polynomial of the principal curvatures, i.e. in an extrinsic way, but on the other hand, it depends only on the intrinsic metric, as on can see from the Brioschi formula, for instance.

Knowing the Gaussian curvature is sufficient to recover its (intrinsic) Riemannian curvature, that is the curvature if the (intrinsic) covariant derivative (in classical geometry of surfaces the ambient Riemannian metric is usually Euclidean, and the ambient curvature is zero).

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