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Differential Geometry
Sample: Section 2.10
Eberhard Malkowsky and Vesna Veliˇ
ckovi´
c
Malkowsky
Mathematisches Institut
Justus–Liebig Universit¨at Gießen
Arndtstraße 2
D-35392 Gießen
Germany
Malkowsky, Veliˇckovi´c
Department of Mathematics
Faculty of Sciences
and Mathematics
University of Niˇs
Viˇsegradska 33
18000 Niˇs, Serbia and Montenegro
email: [email protected]
[email protected]
[email protected]
1
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2.10
2.10
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The Weingarten equations for the derivatives
The Weingarten equations for the derivatives
~ k (k = 1, 2) of the surface
In this section we prove the Weingarten equations for the derivatives. They express the partial derivatives N
~
normal vectors N of a surface in terms of the partial derivatives ~xj (j = 1, 2) of the parametric representation ~x(ui ) of the suface. We also
give a geometric interpretation of the Gaussian curvature.
Theorem 2.86 (The Weingarten equations) Let S be a surface with a parametric representation ~x(ui ) ((u1 , u2 ) ∈ D) and first and
second fundamental coefficients gik and Lik (i, k = 1, 2). We put
Lik = g ij Ljk for i, k = 1, 2.
~ k of the surface normal vectors N
~ of S satisfy the Weingarten equations
Then the partial derivatives N
N~k = −Lik ~xi for k = 1, 2.
(2.111)
~ 2 = 1 implies N
~ •N
~ k (k = 1, 2), the vectors N
~ k are in the tangent plane of the surface, hence we may express them as a
Proof. Since N
linear combination of the vectors ~x1 and ~x2
~ k = Bkm~xm for k = 1, 2.
N
(2.112)
It follows from (2.112) and the definition of the second fundamental coefficients
i
~ k = −g ij Ljk = −Lik for i, k = 1, 2.
Bki = δm
Bkm = g ij gjm Bkm = g ij ~xj • ~xm Bkm = g ij ~xj • N
This implies (2.111).
The Weingarten equations (2.111) reduce when the parameter line of a surface are its lines of curvature.
Theorem 2.87 (The Rodrigues formulae) If the parameter lines of a surface are its lines of curvature, then the Weingarten equations
(2.111) reduce to the Rodrigues formulae
~ k = −κk ~xk for k = 1, 2 where κ1 and κ2 are the principal curvatures.
N
191
(2.113)
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2.10
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The Weingarten equations for the derivatives
Proof. Let S be a surface with a parametric representation ~x(ui ) ((u1 , u2 ) ∈ D). If the parameter lines are the lines of curvature of S,
then it follows from Theorem 2.67 (a) that g12 = L12 = 0, hence
L11 =g 1j Lj1 = g 11 L11 + g 12 L12 =
g22 L11
L11
=
,
g
g11
L21 =g 2j Lj1 = g 21 L12 + g 22 L21 = 0
g12
L22 = 0,
g
g11 L22
L22
=
=
.
g
g22
L12 =g 1j Lj2 = g 11 L12 + g 12 L22 = −
and
L22 =g 2j Lj2 = g 21 L12 + g 22 L22
L11
g11
and
κ2 =
~ 1 = −Lj1~xj = − L11 ~x1 = −κ1~x1
N
g11
and
~ 2 = −Lj2~xj = − L22 ~x2 = −κ2~x2 ,
N
g22
Since the principal curvatures now satisfy
κ1 =
L22
,
g22
we obtain for the Weingarten equations (2.111)
that is the Rodrigues formulae (2.113) hold.
(2.114)
The next result gives a relation between the torsion of an asymptotic line and the Gaussian curvature.
Theorem 2.88 (Beltrami and Enneper, 1870) Let S be a surface with a parametric representation ~x(ui ) ((u1 , u2 ) ∈ D). Then the
torsion of an asymptotic line which is not a straight line is given by
√
|τ | = −K (Figures 136 and 137).
(2.115)
Proof. First, we observe that K < 0 along the asymptotic line by Remark 2.67 (b).
Let the asymptotic line γ be given by a parametric representation ~x(s) = ~x(ui (s)) where s is the arc length along γ, and ~vk (k = 1, 2)
denote the vectors of the trihedra along γ. Since γ has non–vanishing curvature, it follows from Theorem 2.68 that
~ = ±~v3 , hence N
~˙ = ±~v˙ 3 = ∓τ~v2 by Frenet’s third formula (1.29).
N
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2.10
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The Weingarten equations for the derivatives
Using the Weingarten equations (2.111), we obtain
~˙
~˙
~˙
~ × ~x˙ ) = N
~ k • (N
~ × ~x˙ )u˙ k = Lik ~xi • (N
~ • (~xj × ~xi )u˙ k u˙ j ~ × ~xj )u˙ k u˙ j = Lik N
• ~v2 = N
• (~v3 × ~v1 ) = N
• (N
|τ | = N
√ √ = g L2k u˙ k u˙ 1 − L1k u˙ k u˙ 2 = g g 2j Ljk u˙ k u˙ 1 − g 1j Ljk u˙ k u˙ 2 = g 12 L1k u˙ k + g 22 L2k u˙ k u˙ 1 − g 11 L1k u˙ k + g 12 L2k u˙ k u˙ 2 √ = g L1k u˙ k g 12 u˙ 1 − g 11 u˙ 2 + L2k u˙ k g 22 u˙ 1 − g 12 u˙ 2 .
(2.116)
Since the asymptotic line satisfies the differential equation
Lik u˙ i u˙ k = L11 (u˙ 1 )2 + 2L12 u˙ 1 u˙ 2 + L22 (u˙ 2 )2 = 0,
we obtain
L1k u˙ k
hence
2
2
2
2
2
2
= L11 u˙ 1 + L12 u˙ 2 = L211 u˙ 1 + 2L11 L12 u˙ 1 u˙ 2 + L212 u˙ 2 = L11 L11 u˙ 1 + 2L12 u˙ 1 u˙ 2 + L212 u˙ 2
2
2
2
= −L11 L22 u˙ 2 + L212 u˙ 2 = −L u˙ 2
√
√
L1k u˙ k = −L|u˙ 2 |, and similarly L2k u˙ k = −L|u˙ 1 |.
But
L1k u˙ k u˙ 1 + L12 u˙ k u˙ 2 = Lik u˙ i u˙ k = 0
implies
√
L1k u˙ k = ± −L u˙ 2
and
√
L2k u˙ k = ∓ −L u˙ 1
(2.117)
Substituting (2.117) in (2.116) and using gik u˙ i u˙ k = 1, since s is the arc length along the asymptotic line, we obtain
s
√ √
√
√
g −L 2
−L 1
2 1
2
1
1 2
1 2
2 2
=
|τ | =
u˙ −g12 u˙ − g22 u˙ − u˙ g11 u˙ + g12 u˙
+ 2g12 u˙ u˙ + g22 u˙ = −K gik u˙ i u˙ k = −K.
g11 u˙
g
g
This shows (2.115).
Some useful relations can be derived from the Weingarten equations (2.111).
193
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2.10
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The Weingarten equations for the derivatives
Figure 136: The torsion along an asymptotic line on a catenoid an a pseudo–sphere
asymptotic line:
surface normal vectors :
ruled surface RulS :
torsion as a curve on RulS :
194
~y (u1 )
~ (u1 )
N
~ (u1 )
~x(ui ) = ~y (u1 ) + u2 N
~ (t)
~x(t) = ~y (t) + τ (t)N
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2.10
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The Weingarten equations for the derivatives
Figure 137: The ± torsion along an asymptotic line on an explicit surface
Representation as in Figure 136
195
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2.10
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The Weingarten equations for the derivatives
~ of S satisfy
Theorem 2.89 Let S be a surface with a parametric representation ~x(ui ) ((u1 , u2 ) ∈ D). Then the surface normal vectors N
the relation
~2 = N
~ K √g;
~1 × N
N
(2.118)
furthermore, we have
Lki Lkj − Lkj Lki = 0 for i, j = 1, 2.
(2.119)
i
If γ is a curve on S given by a parametric representation ~x(u (s)) where s denotes the arc length along γ, then the surface normal vectors
~ along γ satisfy the relation
N
~˙ )2 = (2HLik − Kgik ) u˙ i u˙ k where K and H are the Gaussian and mean curvature along γ.
(N
(2.120)
Proof. We apply the Weingarten equations (2.111) and obtain
~1 × N
~ 2 = L`1 Lj2 (~x` × ~xj ) = L11 L22 − L21 L12 √g N
~ = g 1m Lm1 g 2n Ln2 − g 2i Li1 g 1k Lk2 √g N
~
N
√
~
gN
= g 11 L11 g 21 L12 + g 11 L11 g 22 L22 + g 12 L21 g 21 L12 + g 12 L21 g 22 L22 − g 21 L11 g 11 L12 + g 21 L11 g 12 L22 + g 22 L21 g 11 L12 + g 22 L21 g 12 L22
√
1
2
2
~
= 2 g11 g22 L11 L22 + g12
gN
L212 − g12
L11 L22 − g11 g22 L212
g
√
g
2
2
2
~ = √g L N
~ = K √g N
~.
N
= g11 g22 L11 L22 − L12 − g12 L11 L22 − L12
2
g
g
Thus we have shown that (2.118) holds.
Furthermore, using the definition of Lki and renaming the indices of summation l and k in the second term to k and m, we obtain
Lki Lkj − Lkj Lki = g km Lmi Lkj − g k` L`j Lki = g km Lmi Lkj − g `k Lki L`j = g km Lmi Lkj − g km Lmi Lkj = 0 for i, j = 1, 2.
Thus we have shown (2.119).
Applying the Weingarten equations (2.111), using the definition of Lki again and observing (2.58) in Remark 2.3.5 (b), we obtain
~1 • N
~ 1 = L`1 Lj1 g`j = g `m Lm1 g jn Ln1 g`j = δjm Lm1 g jn Ln1 = Lj1 g jn Ln1 = L11 g 1n Ln1 + L21 g 2n Ln1
N
= L11 g in Lni + L21 g 2n Ln1 − L11 g 2n Ln2 = 2L11 H + g 2n (L12 Ln1 − L11 Ln2 ) = 2L11 H + g 21 (L12 L11 − L11 L12 ) + g 22 (L12 L21 − L11 L22 )
L
= 2L11 H − g 22 L = 2L11 H − g11 = 2L11 H − g11 K,
g
196
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2.10
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The Weingarten equations for the derivatives
and similarly
~1 • N
~ 2 = Lj1 g jn Ln2 = L12 g 2n Ln2 + L11 g 1n Ln2 = L12 g in Lni + g 1n (−L12 Ln1 + L11 Ln2 )
N
g12
L = 2L12 H − g12 K
= 2L12 H + g 12 L11 L22 − L212 = 2L12 H −
g
and
~2 • N
~ 2 = 2L22 H − g22 K.
N
Thus we have shown
~i • N
~ k = 2HLik − Kgik for i, k = 1, 2.
N
This implies
~˙ 2 = N
~i • N
~ k u˙ i u˙ k = (2HLik − Kgik ) u˙ i u˙ k
N
which is (2.120).
As an application of our results, we consider so–called parallel surfaces, which sometimes can be used to construct surfaces of a given
Gaussian or mean curvature.
Example 2.90 Let S be a surface with a parametric representation ~x(ui ) ((u1 , u2 ) ∈ D) and a ∈ IR be a constant. Then the surface S ∗ with
a parametric representation
~ (ui )
~x ∗ (ui ) = ~x(ui ) + aN
~ denotes the surface normal vector of S
where N
~ k (k = 1, 2) by (2.118) in Theorem 2.89 and the Weingarten
is called a parallel surface (cf. Figure 138). It follows from ~xk∗ = ~xk + aN
equations (2.111) that
~ 1 ) × (~x2 + aN
~ 2 ) = ~x1 × ~x2 + a(~x1 × N
~2 + N
~ 1 × ~x2 ) + a2 (N
~1 × N
~ 2)
~x1∗ × ~x2∗ = (~x1 + aN
√ ~
= ~x1 × ~x2 − a Lk2 ~x1 × ~xk + Lk1 ~xk × ~x2 + a2 K g N
= ~x1 × ~x2 1 + a2 K − a L22 + L11
= (~x1 × ~x2 ) 1 + a2 K − ag ik Lik = (~x1 × ~x2 ) 1 + a2 K − 2aH .
197
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2.10
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The Weingarten equations for the derivatives
~ ∗ of the parallel surface S ∗
Therefore we have for the surface normal vector N
~ ∗ = ±N
~ if 1 − 2aH + a2 K 6= 0.
N
(2.121)
We observe that (2.121) means that if S ∗ is a parallel surface of S, then S is also a parallel surface of S ∗ . It also follows that
√ ∗
√
g = k~x1∗ × ~x2∗ k = g|1 − 2aH + a2 K|.
If |a| is sufficiently small then 1 − 2aH + a2 K > 0, and we obtain for the Gaussian curvature K ∗ of the parallel surface S ∗ by (2.118)
√
gK
1 ~ ∗ ~ ∗ ~ ∗
K
1 ~
∗
~
~
K = √ ∗ N • N1 × N2 = √ ∗ N • (N1 × N2 ) = √ ∗ =
.
(2.122)
g
g
g
1 − 2aH + a2 K
Since S is a parallel surface of S ∗ , we obtain by interchanging the roles the Gaussian and mean curvature of S and S ∗ and replacing a by
−a
K∗
.
K=
1 + 2aH ∗ + a2 K ∗
This implies
1 − a2 K
K
∗
∗
2 ∗
∗
2
2aH K = K − a K K − K = K (1 − a K) − K = K
−1 =
(1 − a2 K − 1 + 2aH − a2 K),
2
1 − 2aH + a K
1 − 2aH + a2 K
and we obtain for K 6= 0
H − aK
.
(2.123)
1 − 2aH + a2 K
Equations (2.122) and (2.123) may be used to find surfaces with given Gaussian or mean curvature.
First we consider the surface of rotation RS with
1
1
Z s
π π
u
λ2
u
1
1
2
r(u ) = λ cos
and h(u ) =
1 − 2 sin
du1 for u1 ∈ I1 ⊂ − ,
where λ and c are constants with 0 < λ < c.
c
c
c
2 2
H∗ =
198
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2.10
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The Weingarten equations for the derivatives
Then we have
λ
r0 (u1 ) = − sin
c
s
1
u1
λ2
r(u1 )
u
0 1
2
, h (u ) = 1 − 2 sin
, (r0 (u1 ))2 + (h0 (u1 ))2 = 1 and r00 (u1 ) = − 2 ,
c
c
c
c
and the Gaussian curvature of RS is given by (2.64)
r00 (u1 )
1
= 2.
1
r(u )
c
2
Thus RS has constant Gaussian curvature K = 1/c . If we choose a = c, then it follows from (2.123) for the mean curvature H ∗ of the
parallel surface RS ∗ for a of RS
1
H−
c = − 1 1 − cH = − 1 ,
H∗ =
2(1 − cH)
2c 1 − cH
2c
∗
∗
that is RS has constant mean curvature H = −1/(2c) (cf. Figure 139). The same argument holds true if we choose RS as above, but
with λ > c and u1 ∈ (−c arcsin (c/λ), c arcsin (c/λ)).
Now we consider the pseudo–sphere which has constant Gaussian curvature K = −1 by (2.65) in Example 2.38. If we choose a = 1, then
it follows from (2.122) and (2.65) for the Gaussian curvature K ∗ of the parallel surface RS ∗ for a of RS
p
exp (−u1 ) 1 − exp (−2u1 )
1
1
∗
=
=
(cf. Figure 140.)
K =−
1 − 2H − 1
2H
1 − 2 exp (−2u1 )
K(u1 ) = −
Now we give a geometric interpretation of the Gaussian curvature. The following definition is important.
Definition 2.91 Let S be a surface with a parametric representation ~x(ui ) ((u1 , u2 ) ∈ D). The map that assigns to every point P of S the
surface normal vector at P is called the spherical Gauss map (cf. Figures 141 and 142).
Let S be a surface with a parametric representation ~x(ui ) ((u1 , u2 ) ∈ D), P ∈ S be a point given by the parameters u10 and u20 , and U be
a neighbourhood of P , given by a subset DU of D. The image of U under the Gauss map defines a part U ∗ of the unit sphere. If AU and
AU ∗ denote the surface areas of U and U ∗ –including their signs –, then it follows that
RR
~ (ui ) • N
~ 1 (ui ) × N
~ 2 (ui ) du1 du2
N
Du
AU ∗
= RR
.
(2.124)
~ (ui ) • (~x1 (ui ) × ~x2 (ui )) du1 du2
AU
N
Du
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2.10
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The Weingarten equations for the derivatives
Figure 138: A catenoid and a parallel surface
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2.10
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The Weingarten equations for the derivatives
Figure 139: Surfaces of constant Gaussian curvature K = 1 and their parallel surfaces with constant mean curvature H = −1/2
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2.10
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The Weingarten equations for the derivatives
Figure 140: A pseudo–sphere and its parallel surface for a = 1
202
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2.10
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The Weingarten equations for the derivatives
Figure 141: A curve on a monkey saddle and its image under the spherical Gauss map
203
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2.10
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The Weingarten equations for the derivatives
Figure 142: A curve on a conoid and its image under the spherical Gauss map
204
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2.10
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The Weingarten equations for the derivatives
The ratio in (2.124) is the bigger, the more the surface S is curved, since then the surface normal vectors cover a larger area of the unit
sphere (cf. Figure 143). If the neighbourhood U of P shrinks to the point P , that is if we take the limit U → {P } in (2.124), then we
obtain, using (2.118) in Theorem 2.89
p
~ (ui0 ) • N
~ 1 (ui0 ) × N
~ 2 (ui0 )
N
~ (ui0 ) • N
~ (ui0 ) g(ui )K(ui0 )
N
AU ∗
p
= K(ui0 ).
lim
=
=
(2.125)
i
i
i
i
~
U →{P } AU
g(u0 )
N (u0 ) • (~x1 (u0 ) × ~x2 (u0 ))
The value in (2.125) only depends on the point P on S and was originally introduced by Gauss as the measure of the curvature of a surface
at a point. The sign of K which determines the elliptic, parabolic or hyperbolic property of P has the following geometric significance: It
is positive or negative depending on whether the Gauss map preserves or reverses the orientation (cf Figure 144).
Example 2.92 We verify (2.125) without the use of the above results in the case of the torus with a parametric representation
~x(ui ) = {(r0 + r1 cos u1 ) cos u2 , (r0 + r1 cos u1 ) sin u2 , r1 sin u1 } ((u1 , u2 ) ∈ D = (0, 2π)2 ) where r0 and r1 are constants with 0 < r1 < r0 .
Furthermore, we choose a point P0 on the torus given by the parameters (u10 , u20 ) ∈ D and a neighbourhood U of P defined by the rectangle
R = [u10 − w1 , u10 + w1 ] × [u20 − w2 , u20 + w2 ] where w1 and w2 are positive reals and R ⊂ D. Then the surface area of U is
AU =
ZZ p
U
g(ui ) du1 du2 =
1 +w 1 u2 +w 2
uZ
0
0
Z
r1 (r0 + r1 cos u1 ) du1 du2 = 2r1 w2
u10 −w1 u20 −w2
1 +w 1
uZ
0
(r0 + r1 cos u1 ) du1
u1 −w1
0
1
1
1
1
1 2
1 2
2 2
= 4r0 r1 w w + 2r1 w sin (u0 + w ) − sin (u0 − w ) = 4r0 r1 w w + 4r12 w2 sin w1 cos u10
= 4r1 w2 r0 w1 + r1 sin w1 cos u10 .
A straightforward computation shows that the surface normal vectors of the torus and its partial derivatives are given by
~ (ui ) = −{cos u1 cos u2 , cos u1 sin u2 , sin u1 },
N
~ 2 (ui ) = {cos u1 sin u2 , − cos u1 cos u2 , 0}
N
hence
~ 1 (ui ) = {sin u1 cos u2 , sin u1 sin u1 , − cos u1 },
N
~ (ui ) • (N
~ 1 (ui ) × N
~ 2 (ui )) = cos u1 .
N
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2.10
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The Weingarten equations for the derivatives
Figure 143: The spherical images of the same domain of two different paraboloids
206
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2.10
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The Weingarten equations for the derivatives
Figure 144: Curves on a torus and their orientations in the neighbourhoods of points with K > 0, K = 0 and K < 0, and their images
under the Gauss map
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2.10
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The Weingarten equations for the derivatives
Figure 145: Neighbourhoods of an elliptic and hyperbolic point on a torus
208
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LIST OF FIGURES
.
LIST OF FIGURES
This implies
ZZ
AU ∗ =
~ (ui ) • (N
~ 1 (ui ) × N
~ 2 (ui )) du1 du2 =
N
U
1 +w 1 u2 +w 2
uZ
0
0
Z
cos u1 du1 du2 = 2w2 sin (u10 + w1 ) − sin (u10 − w1 ) = 4w2 cos u10 sin w1 ,
u10 −w1 u20 −w2
and so
AU ∗
=
AU
2
4w cos u10 sin w1
4r1 w2 (r0 w1 + r1 cos u10 sin w1 )
sin w1
cos u10
w1
→
= = K(u10 )
1
1
sin
w
r
(r
+
r
cos
u
)
1
0
1
0
r1 r0 + r1 cos u10
w1
cos u10
((w1 , w2 ) → (0, 0)),
by (2.94) in Example 2.52 (b).
List of Figures
136
137
138
139
140
141
142
143
144
The torsion along an asymptotic line on a catenoid an a pseudo–sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The ± torsion along an asymptotic line on an explicit surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A catenoid and a parallel surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Surfaces of constant Gaussian curvature K = 1 and their parallel surfaces with constant mean curvature H = −1/2 . . . .
A pseudo–sphere and its parallel surface for a = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A curve on a monkey saddle and its image under the spherical Gauss map . . . . . . . . . . . . . . . . . . . . . . . . . . .
A curve on a conoid and its image under the spherical Gauss map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The spherical images of the same domain of two different paraboloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Curves on a torus and their orientations in the neighbourhoods of points with K > 0, K = 0 and K < 0, and their images
under the Gauss map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
145 Neighbourhoods of an elliptic and hyperbolic point on a torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
209
194
195
200
201
202
203
204
206
207
208