0
$\begingroup$

It is the first time that I meet the concept "normal curvature matrix".

By google searching, it seems like a concept in differential geometry. But I know little of differential geometry and do not get any clear definition, properties of it or information on how to calculate it.

Is there any references or ideas on how to understand it?

1)Google books: Google books link

2)Springer book: Springer book link

The content in the second link made me believe this should be a concept in differential geometry.

$\endgroup$
4
  • $\begingroup$ Most likely it is differential geometry, but for better understanding it would be better to describe the context. $\endgroup$ Feb 5, 2014 at 16:43
  • $\begingroup$ As a differential geometer, I have not come across this term, and neither has Google. What is the source where you found the term used? I am guessing that it refers to the curvature of the canonical connection on the normal bundle of a submanifold of a Riemannian manifold. $\endgroup$
    – Ben McKay
    Feb 5, 2014 at 19:52
  • $\begingroup$ @BenMcKay you should use "normal curvature matrix" with the quotation marks for google search; I have updated it with the two links I found. $\endgroup$ Feb 6, 2014 at 1:00
  • $\begingroup$ @AlexDegtyarev the concept would be that in the first link. It is used to determine whether a numerical solution $x^*$ is local minimum or not. Since there is already positive definite Hessian which also requires second order derivative information, what is the advantage of using the maximum eigen value of this normal curvature matrix? $\endgroup$ Feb 6, 2014 at 1:05

1 Answer 1

1
$\begingroup$

You're looking at the Weingarten map, or shape operator. This can be seen as a map from the tangent space of the surface to itself. It takes a tangent vector $v$ to the derivative of the normal in the direction $v$.

$\endgroup$
1
  • $\begingroup$ I want to understand how this concept can be used for the local minimum criterion in unconstrained nonlinear optimization, the proof and its advantage over Hessian (positive semi-definiteness). Any suggestions? thank you! $\endgroup$ Feb 6, 2014 at 1:43

Not the answer you're looking for? Browse other questions tagged or ask your own question.