An example of the first/second fundamental form

Question

I have some trouble understanding the first/second fundamental form, so I guess a worked-out example would really help.

Let's say for the graph of a function $g(x,y)$ with respect to the natural chart. What are the matrices for the first fundamental form, the second fundamental form, the differential of the Gauss map, and the Gaussian curvature?

Thomas · Accepted Answer · 2016-04-01T07:18:59.333

2

The natural parametrization is $F(x,y)= (x,y,g(x,y))$.

So $F'_x=(1,0,g'_x) ; F'_y=(0,1, g'_y)$.

Then ${ds}^2= (1+{g'_x}^2){dx}^2$+$2g'_xg'_y dx dy+(1+{g'_y}^2 ){dy}^2$.

A normal vector is $F'_x \wedge F'_y=(-g'_x,-g'_y,1)$,

The Gauss map is $N(x,y)=-{1\over \sqrt {1+{g'_x}^2+{g'_x}^2}}( g'_x, g'_y,-1)$.

The second form is obtained while deriving $N(x,y)$, which yields an ugly formula.

The case where $g'_x(0,0)=g'_y(0,0)=0$ (ie the surface is horizontal) is nice and easy, the second form is given by a immediate computation $ g''(0,0)= -Hess(g)(0,0)$. Note the sign -. For instance the hemi-sphere $z= \sqrt {R^2-x^2-y^2}= R-{x^2+y^2\over 2R}+o(x^2+y^2)$ has positive curvature $1/R^2$, as $g''(0,0)=-{1\over R} Id$.

edited Apr 01 '16 at 07:18

answered Mar 30 '16 at 13:11

Thomas

7,470

I guess the Gauss map should be $N(x,y)=\frac{( g'_x, g'_y,-1)}{-\sqrt {1+{g'_x}^2+{g'_x}^2}}$? – adrw_k Mar 31 '16 at 14:24
Yes, a typo, sorry. – Thomas Mar 31 '16 at 18:12

An example of the first/second fundamental form

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