Part 17
Shape Operators of Surfaces in R3
Printed version of the lecture Differential Geometry on 6. May 2010
Tommy R. Jensen, Department of Mathematics, KNU
17.1
Overview
Contents
1
The Shape Operator
1
2
Examples
4
3
Conclusion
6
17.2
1
The Shape Operator
Euclidean vector fields
Let M be a connected surface in R3 .
Definition
A Euclidean vector field on M is a function Z that maps every point p of M to a tangent vector Z(p) to R3
at p.
Definition
Let Z be a Euclidean vector field on M.
Let v be a tangent vector to M at a point p. So v ∈ Tp (M).
The covariant derivative of Z with respect to v is the tangent vector
∇v Z = Z(α)0 (0)
to R3 at the point p, where α is a curve in M of initial velocity α 0 (0) = v.
Applying this definition directly to calculating ∇v Z is called Method 1.
1
Remark. If Z is a tangent vector field to M and v is a tangent vector, it does not imply that ∇v Z is a
tangent vector to M.
17.3
A different method to calculate ∇v Z
Method 2
Express Z in terms of its Euclidean coordinate functions
Z = ∑ ziUi ,
where {U1 ,U2 ,U3 } is the natural frame field of R3 .
Then
∇v Z = ∑ v[zi ]Ui .
This method is correct, since
∑ v[zi ]Ui = ∑ zi (α)0 (0)Ui = Z(α)0 (0).
17.4
17.5
Unit normal vector fields
Example
If M is orientable, there exists a unit normal vector field U on M.
Since M is connected, there are exactly two such vector fields on M : U and −U.
If M is not orientable, a small region around every point is diffeomorphic to R2 , which is an orientable
surface.
Therefore there always exist unit normal vector fields ±U locally around every point.
2
The directions may not change suddenly to the opposite.
17.6
The shape operator at a point of a surface
Definition 1.1
Let p be a point of M.
Let U be a unit normal vector field defined on an open neighborhood of p in M.
For any tangent vector v to M at p define
S p (v) = −∇vU.
Then S p is the shape operator of M at p derived from U.
The shape operator of M at p derived from −U is obviously equal to −S p .
The shape operator explains how M curves around p.
17.7
Linearity
Lemma 1.2
The shape operator S p at p ∈ M derived from U is a linear function
S p : Tp (M) → Tp (M).
Proof of Lemma 1.2
The vector field U has unit length, and therefore U •U = 1.
If v is a tangent to M at p, then
0 = v[U •U] = 2∇vU •U(p) = −2S p (v) •U(p).
Since U(p) is normal, the tangent vectors to M at p that are orthogonal to U(p) are precisely the elements
of Tp (M).
Therefore S p (v) ∈ Tp (M) follows.
For a, b ∈ R and v, w ∈ Tp (M) we calculate:
S p (av + bw) = −∇av+bwU = −(a∇vU + b∇wU) = aS p (v) + bS p (w).
17.8
3
The shape operator of M
Definition
The shape operator of M is the set
S=
[
{±S p : p ∈ M}.
17.9
2
Examples
Shape operators of some surfaces
Example 1.3(1): the sphere Σ
Let Σ be the sphere of radius r :
Σ = {p ∈ R3 : ||p|| = r}.
Let U be the unit normal vector field that points to the outside.
U=
Then
1
xiUi .
r∑
1
v
v[xi ]Ui (p) = .
r∑
r
v
And we have
S(v) = − .
r
It means that a sphere curves in the same way everywhere.
It curves more when the radius is small than when it is large.
So
∇vU =
17.10
Shape operators of some surfaces
Example 1.3(2): a plane in R3
Let P be any plane in R3 .
A unit normal vector field U on P is a parallel vector field.
It follows that
S(v) = −∇vU = 0.
This means that the plane does not curve in any way.
17.11
4
Shape operators of some surfaces
Example 1.3(3): a circular cylinder in R3
Let C : x2 + y2 = r2 be the cylinder in R3 of all points at distance r from the z-axis.
Let U be the unit normal vector field that points to the outside of C. Then U = (xU1 + yU2 )/r.
The tangent plane Tp (C) to C at a point p = (p1 , p2 , p3 ) is spanned by the tangent vectors e1 = (0, 0, 1) p
and e2 = (−p2 , p1 , 0) p .
We get S(e1 ) = 0 and S(e2 ) = −e2 /r.
It means that C curves in some way like a sphere and in some way like a plane.
17.12
Example 1.3(4): the saddle surface in R3
Let M : z = xy. Let p = (0, 0, 0). Then p ∈ M.
The x-axis {(t, 0, 0) : t ∈ R} and the y-axis {(0,t, 0) : t ∈ R} are curves in M.
Therefore the vectors u1 = (1, 0, 0) p and u2 = (0, 1, 0) p are tangents to M at p.
Hence (0, 0, 1) p is a unit normal tangent vector to M at p.
This defines a unit normal tangent vector field U around p.
We can calculate ∇u1 U = −u2 , and ∇u2 U = −u1 .
Hence S(au1 + bu2 ) = bu1 + au2 .
17.13
Symmetry of the shape operator
Definition
Let V ⊂ Rn be a vector space.
Let F : V → V be any function.
Then F is called symmetric, if
F(v) • w = v • F(w) for all v, w ∈ V .
Lemma 1.4
Let p be any point in M ⊂ R3 .
Then the shape operator S : Tp (M) → Tp (M) is a symmetric linear operator.
The proof of Lemma 1.4 will have to wait until Section 4 on Computational Techniques.
17.14
5
3
Conclusion
The End
17.15
Next time:
Normal Curvature
17.16
6