2.The second fundamental form of regualr surfaces

2.1 Normal Curvature

We have seen the equation 1 and 2, the second fundamental form is defined by?, then consider the basis vector, we have :

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

If the vector??is the velocity vector of a curve parametrized by arclength, then it should be that:?,?, then we have?? ? ? ? ?

? ? ? ? ? ? ? ? ?

The definition of normal curvature is:?, where??is the curvature of the regular curve passing through point?,??is the angle between nomal vector of the surface and normal vector of the curve. Therefore, if the vector is of?,?we have:

? ? ? ? ? ? ??

In other words, the value of the second fundamental form for a unit vector is equal to the normal curvature of a regular curve passing through??and tangent to?. In particular, we obtained the following result.

2.2 Meusnier theorem

?and??have the same normal curvature at??along?.

All curves lying on a surface at a given point??the same tangent line have at this point the same normal curvatures.

Example:

The normal curvature of any point on the unit sphere is 1, because consider a normal tangent vector??of a geodesic, we have the normal vector?,then we know that?, thererfore the normal curvature is that:?

2.3 Principle Direction?

The directions given by the eigenvectors??are called principal directions at?. And donate?,?we have that:

then,??(Gaussian curvature, will be introduced later)

2.2.1?Line of curvature

Defined as?, where??is the tangent vector of a regular curve,donated as?. Therefore, we have:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

for unit vector in tangent space, we can have:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

then the second fundamental form is:

Because?, we can derive:

therefore we conclude that?.

Moreover we have the following properties on any point on the surface,the point is:

• Elliptic if??

• Hyperbolic if??

• Parabolic if???but?

• .Planer if??

2.2.2?Umbilical point

The point with?, the particular condition is that?, the the planar points are umbilical.

we can conclude that if all points are umbilical, then the surface is sphere or plane.

2.2.3?Asymptotic direction

Direction that normal curvture is 0,that is?.

2.3?The second fundamental form

Because the tangent vector is?,?then the second fundamental form is:

then:

the Gaussian curvature and mean curvature are?given by:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

and the relationship between normal curvature, gaussian curvature, mean curvature is:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

By above,we can directly calculate the Gaussian curvature by:

Back to asymptotic direction, if a curve is of 0 normal curvature, then the second fundamental form is also 0,then the differential equation for asymptotic curve is given by:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

for line of curvature, we have the following relationship:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

Consider surface revolution:?,the the first fundamental form is:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Furthermore, if the curve is parametrized by arclength,we have that:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

then the Gauss curvature is given by：

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

When the case?，we have the following :

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

If the surface is parametrized by：,then?we have:??

and?

? ? ? ? ??

3? Gauss theorem and compactibility?

My last blog mentioned Christoffel symbols, here we review:

then we have:

? ? ? ? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ? ? ? ?

in the case F=0, we obtain:

Brioschi Formular

The Gaussian curvature is intrinsic of the surface, that is, can only expressed by the first fundamental form: we have:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

where

in the case of?,?we have the special form:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

4?Isometries; Conformal map

4.1?Isometry

A differomorphism map??is an isometry if for all points and pairwise vectors:?satisfying:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

Consider on?,?the parametrization is?,?and the relationship is?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

we have to calculate?,?with respect to?.?In this case, we have to make sure that if?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

then??is isometry.

4.2 Conformal map

A differomorphism map??should satisfying that:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

The following statements are equivalent:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?