微分幾何復(fù)習(xí)筆記（第二基本形式）（1）
考完了，考的還不錯(cuò)，傳個(gè)復(fù)習(xí)筆記??????????? 有些證明就不證了，反正也沒考到。前幾章比較簡單，雖然也有寫，看心情傳吧′?`
Definition and properties
Consider regular surface ，for any point?
,? the tangent space?
? is spanned by
,then the unit normal vector is??
.
That is, we have a differentiable map?? that associates to each?
?
a unit normal vector??
1.1 Gauss map
Let S in? be a surface with an orientation?
.The map?
? ?takes its values in the unit sphere?
?.
hen, we have the Gauss map? , which is differentiable.We can know that the direction of each N reserves after the Gauss map, which is just remove all normal vectors from the surface?
?to the unit sphere.
Also we get:? .
Then we can consider the curve in? by the mapping?
?from the surface?
?that is?
in?
then the tengent vectors? are mapped to the sphere, that is?
Example 1:
Consider the unit sphere??.A regular on?
?is given by?
, then?
.Therefore we obtain:
Example 2:
Consider the cylinder, that is?. The tangent vectors?
,the unit normal vector inward is?
.
Then we have?.
By comparing with? and?
, we find that:
;?
.
Therefore?? is of 2 eigen values?
.
It is suprising to discover that, the Gauss map maps the points from? to?
, to with different shapes.Consider the plane?
in?
, the normal vector is?
,then the Gauss map of the plane is?
That is, all the points in the plane?
are mapped to the point?
on the unit sphere.
Consider the cylinder, that is??the normal vectors are?
and?
.Therefore we find that all points on the cylinder are mapped to the circle?
?on the unit sphere.
1.2 Weingarten map
Actually the weingarten map is?...
Let a regular curve?on?, then by Gauss map we have?
Then?
In particular,?, that is:
? ? ? ? ? ? ? ? ? ? ??
Just know this concept is ok..
1.3 Self-adjoint linear map
The Weingarten map is self-adjoint, which is of the property:?
Consider basis vector, we have:
Proof:
Because?
;
thus we have:
? ? ? ? ? ? ? ? ? ? ? ? ? ?
Hence Weingarten map is self-adjoint.
In the proof above, we also descover:
The 2 equations above are much more important later?. . .?
Latex 公式在這里居然算圖片，不能超過100條....分開更算了
