Chapter 4: Geometry

8. The principal directions Ù Ú are obtained by solving the homogeneous
equation,
½« ¾¬ Ù « Ù ¬ ¾« ½¬ Ù « Ù ¬ ¼ (4.20.14)
or
´ µ Ù ¾ ·´ µ Ù Ú ·´ µ Ú ¾ ¼ (4.20.15)
9. Rodrigues formula: Ù Ú is a principal direction with principal curvature
if, and only if, N · x ¼.
10. A point x f´Ù Úµ on Ë is an umbilical point if and only if there exists a
constant such that «¬´Ù Úµ «¬´Ù Úµ.
11. The principal directions at x are orthogonal if x is not an umbilical point.
12. The Ù- and Ú-parameter curves at any non-umbilical point x are tangent to
the principal directions if and only if ´Ù Úµ ´Ù Úµ ¼. If f defines a
coordinate patch without umbilical points, the Ù- and Ú-parameter curves are
lines of curvature if and only if ¼.
13. If ¼ on a coordinate patch, the principal curvatures are given by
½ , ¾ . It follows that the Gaussian and mean curvatures have
the forms
Ã
14. The Gauss equation: x «¬
and À ½ ¾
­
«¬ x ­ · «¬ n.
15. The Weingarten equation: n « «¬ ¬­ x ­ .
·
(4.20.16)
16. The Gauss–Mainardi–Codazzi equations: «¬ ­Æ «­ ¬Æ Ê Æ«¬­ , «¬­
Æ
«­¬ · «¬ Æ
Æ­ «­ Ƭ ¼, where Ê Æ«¬­ denotes the Riemann curvature
tensor defined in Section 5.10.3.
THEOREM 4.20.2
(Gauss’s theorema egregium)
The Gaussian curvature à depends only on the components of the r st fundamental
metric «¬ and their derivatives.
THEOREM 4.20.3
(Fundamental theorem of surface theory)
If «¬ and «¬ are suf ciently differentiable functions of Ù and Ú which satisfy the
Gauss–Mainardi–Codazzi equations, Ø´ «¬ µ ¼, ½½ ¼, and ¾¾ ¼, then a
surface exists with Á «¬ Ù « Ù ¬ and ÁÁ «¬ Ù « Ù ¬ as its r st and second
fundamental forms. This surface is unique up to a congruence.
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