c*******d
发帖数: 353 | 1 In the book 'differential geometry' by Kreyszig, a result is frequently used
about the principal curvature k1, k2. For example, we know that gaussian
curvature K=k1*k2.
When lines of curvature (curves with principal curvature as tangents)
coincide with coordinate curves, it can be shown k1 = b_1^1, the first
element of a mixed tensor with degree 2 and 1 covariance indice. (p.131)
The author then equate k1 = b_11/g_11, k2=b_22/g_22. And this result is used
in several places. Here is what I am hav | c*******d
发帖数: 353 | | s*********l
发帖数: 103 | 3 The result in the book is correct.
It seems a special case of Weingarten Equations.
http://mathworld.wolfram.com/WeingartenEquations.html
Make sure your computation of g^cv is correct.
(is it the inverse of the matrix form of the first
fundamental form?)
【在 c*******d 的大作中提到】
: 难道没有人知道吗？
| c*******d
发帖数: 353 | 4 ah, I found a small glitch in my calculation, now it all makes sense
b_1^1 = b_11 g^11 = b_11 g_22/g = b_11 g_22/(g_11 g_22) = b_11 / g_11
thanks |
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