# Extrinsic intrinsic

1. Feb 8, 2010

### wofsy

The Gauss curvature of a surface in R^3 is intrinsic i.e. it is an invariant of local isometry.

For a hyper-surface of R^n is this also true? By this I mean: The Gauss curvature is the determinant of the Gauss mapping of the surface into the unit sphere. Is the determinant of the Gauss mapping of a hypersurface in higher dimensions also intrinsic?

I am not sure if you can restate this question as follows.

Is the connection on a 2d surface induced by the Gauss mapping? i.e. is it the pull back of the standard connection on the unit sphere under the Gauss mapping? If so this would make the Gauss curvature intrinsic. Same question about connections for higher dimensions.

Last edited: Feb 9, 2010
2. Feb 8, 2010

### Ben Niehoff

On a 2-surface, the Ricci scalar is equal to twice the Gauss curvature, so yes, it is an intrinsic quantity in general.

I'm not sure how you define Gauss curvature for higher-dimensional surfaces. Of course, the Ricci scalar will always be an intrinsic quantity, regardless. I think the Ricci tensor represents sectional Gaussian curvature; i.e., plug in two vectors, and the result Ric(u,v) gives you the local Gaussian curvature on the 2-dimensional surface spanned by u and v. I could be wrong on that, though.