Dynamics at the Horsetooth Volume 1, 2009.
The geometry of a class of partial differential equations and surfaces in
Lie algebras
Luke Bayens
Department of Mathematics
Colorado State University
bayens@math.colostate.edu
Report submitted to Prof. P. Shipman for Math 540, Fall 2009
Abstract. We show how an immersion of a surface in R3 leads to a set of partial
differential equations, and how these equations are related to the study of surfaces in
Lie groups and Lie algebras.
Keywords: Riemannian geometry, surface, Lie group, Lie algebra
1
Introduction
The most important work in the history of differential geometry is Gauss’ 1828 paper Disquisitiones
generales circa superficies curvas [5]. In this work Gauss describes properties of surfaces embedded
in R3 , in particular the so called Gaussian curvature. He also describes a system of equations which
over the course of time have proven to be fundamental in the analysis of surfaces; indeed, the Gauss
system and the symmetries it admits for certain classes of surfaces underpin a remarkable connection
between classical differential geometry and modern work in physics, differential equations and
algebra.
There is a wealth of material in the literature discussing the links between the differential
equations arising from the geometry and physical systems (for example, the Korteweg de-Vries
equation and how it relates to long wave propogation and the theory of solitons). There are many
significant resources devoted to analyzing the solutions of these equations, superposition principles,
numerical methods for finding solutions, and so forth. However, the author found it difficult to
find a satisfactory exposition of how the differential equations arise from the geometry. There is an
unfortunate clash of style, standard of rigor, notation and emphasis between the pure differential
geometry works and their applied counterparts. To the differential geometer, embedding surfaces
in R3 is passé, and in any case, he is not particularly interested in the differential equations that
arise. The applied mathematician is interested in studying these equations and their physical
implications, but not usually interested in the formalism of the geometry.
This paper tries to address this situation by presenting the classical theory of surfaces from a
modern perspective, with particular emphasis given to the resulting differential equations. In the
final section we hint at a relationship between surfaces in R3 and surfaces in Lie groups and Lie
algebras.
PDEs and surfaces in Lie algebras
2
Luke Bayens
Elements of the theory of surfaces in R3
We assume that the reader has a familiarity with the basics of Riemannian manifolds. In particular,
an understanding of curvature would be helpful. For an excellent overview of the area see [7, 6, 3].
Consider an embedded 2-manifold M ⊂ R3 , with i : M → R3 the inclusion map. The
first fundamental form I is I = i∗ h , i, where h , i is the usual Riemannian metric on R3 .
(Throughout this paper we use subscript asterisk to denote the pushforward or differential of a
map, and superscript asterisk to denote the pullback of a map.) In terms of a coordinate system
χ = (x, y) on M , we can write the tensor I on M as
I = E dx ⊗ dx + F dx ⊗ dy + F dy ⊗ dx + G dy ⊗ dy
for some functions E, F , G on M . If the inverse of χ is r : U → R3 for some open U ⊂ R2 , then
E(r(u, v)) = hr1 (u, v), r1 (u, v)i
F (r(u, v)) = hr1 (u, v), r2 (u, v)i
G(r(u, v)) = hr2 (u, v), r2 (u, v)i,
where ri denotes the partial derivative of r with respect to the ith variable. Here we see our
first abuse of notation – when we write ri (u, v) we are thinking of ri (u, v) ∈ Tr(u,v) M which we
are identifying with Tr(u,v) R3 . We should really write ri (u, v)r(u,v) ∈ Tr(u,v) M ' Tr(u,v) R3 or
something similar, but for obvious reasons we will not1 . Since E = hr1 , r1 i ◦ r−1 , and so on, this
sometimes makes the functions E, F, G rather awkward to work with; it is often more convenient
to define everything in terms of a given immersion.
If r : M → R3 is an immersion, the first fundamental form Ir of r is defined to be the tensor
∗
r h , i on M . In particular, when r : U → R3 for open U ⊂ R2 we have a form Ir on U defined by
Ir (u, v)(X1 , X2 ) = hr∗ X1 , r∗ X2 i,
for X1 , X2 ∈ T(u,v) R2 .
We can then define functions E, F, G on U by
E = hr1 , r1 i,
F = hr1 , r2 i,
G = hr2 , r2 i.
These are nothing more than the components of Ir = r∗ h , i with respect to the standard
coordinate system (u, v) on R2 . We introduce the subscript notation gij = hri , rj i, so that
g11 = E,
g12 = g21 = F, g22 = G,
P
ij
∗
and the superscript notation g ij such that k gij g kj = δij , that is, (gij )−1
√ = (g ). Since r h , i is
2
positive definite, it follows that det(gij ) = EG − F > 0 and |r1 × r2 | = EG − F 2 .
For every p ∈ M , we can consider the tangent space Tp M as a subspace of Tp R3 by identifying
Tp M with i∗ (Tp M ) ⊂ Ti(p) R3 = Tp R3 . In the vector space Tp R3 , with inner product h , ip , the
subspace Tp M has an orthogonal complement (Tp M )⊥ ⊂ Tp R3 , and we can use the decomposition
Tp R3 = Tp M ⊕ (Tp M )⊥ to write any v ∈ Tp R3 as v = v T + v N for v T ∈ Tp M (the tangential
projection of v) and v N ∈ (Tp M )⊥ (the normal projection of v). If M is oriented, then on a
neighborhood U of p ∈ M there is a unique vector field ν on M such that hν, νi = 1, ν(q) ∈ (Tp M )⊥ ,
and (X1 , X2 ) ∈ Tp M is positively oriented if and only if (X1 , X2 , ν(p)) is positively oriented in R3 .
We will usually regard ν : M → S 2 ⊂ R3 and write this normal vector field in the very abusive
form
p 7→ ν(p)p ∈ Tp M.
1
It is the author’s opinion that while abuse of notation is a necessary part of advanced mathematics,
mathematicians should at least be aware when it is happening.
Dynamics at the Horsetooth
2
Vol. 1, 2009
PDEs and surfaces in Lie algebras
Luke Bayens
If instead we are dealing with an immersion r : M → R3 , the normal field should be considered
as a vector field along r, since we may have points p,q ∈ M with r(p) = r(q), but with different
normals at this point. In this case we denote the normal vector field along r by
q 7→ N (q)r(q) ∈ Tr(q) R3 .
Here N is a function N : M → S 2 ⊂ R3 . We will adopt the convention of using ν when considering
embedded submanifolds M ⊂ R3 and N when considering immersions r : M → R3 . If W ⊂ M is
an open set on which r is an embedding, then a unit normal field ν on r(W ) ⊂ R3 is determined by
the condition that N = ν ◦ f on W . In terms of ν we can define the second fundamental form
II on M by
II(p)(X1 , X2 ) = h−ν∗ X1 , X2 i, for X1 , X2 ∈ Tp M.
This is once again an abuse of notation; we are identifying2 Tν(p) R3 ' Tp R3 and thinking
of ν∗ : Tp M → Tp M so that the inner product nmakes sense.
o Notice that the matrix of
−ν∗ : Tp M → Tp M with respect to the ordered basis (r1 )p , (r2 )p is (gij )−1 · (`ij ).
Similarly, we define the second fundamental form IIr of r to be the tensor on M defined by
IIr (q)(X1 , X2 ) = h−N∗ X1 , r∗ X2 i,
for X1 , X2 ∈ Tq M,
with the same identifications. Equivalently, we have IIr = r∗ II.
In particular, consider an immersion r : U → R3 , for U ⊂ R2 open. We can choose N explicitly
to be
r1 × r2
r1 × r2
N=
.
=√
|r1 × r2 |
EG − F 2
Then
IIr (u, v)(X1 , X2 ) = h−N∗ X1 , r∗ X2 i,
for X1 , X2 ∈ T(u,v) R2 .
We can define the functions e, f, g on U by
e = h−N1 , r1 i = hN, r11 i,
f = h−N1 , r2 i = hN, r12 i,
g = h−N2 , r2 i = hN, r22 i.
Thus e, f , g are simply the components of IIr with respect to the standard coordinate system (u, v)
on R2 . Again, it is sometimes convenient to use the subscript notation
`ij = h−Ni , rj i = hN, rij i.
At this pointnwe can see osome use for all this notation. The matrix M of −ν∗ : Tp M → Tp M
with respect to (r1 )p , (r2 )p is
M = (gij )−1 · (`ij )
1
G −F
e f
=
E
f g
EG − F 2 −F
with ri , gij , `ij are evaluated at (u, v), where p = r(u, v). The principle curvatures k1 , k2 are the
eigenvalues of M , the Gaussian curvature K is k1 · k2 , and the mean curvature H is (k1 + k2 )/2.
So K and H are the determinant and half the trace, respectively, of M . Thus,
ef − g 2
EG − F 2
Eg − 2F f + Ge
H=
,
2(EG − F 2 )
K=
2
Here we are using the Levi-Civita connection of the standard Riemannian metric on R3 .
Dynamics at the Horsetooth
3
Vol. 1, 2009
PDEs and surfaces in Lie algebras
Luke Bayens
where the left hand sides must be evaluated at (u, v) when the right hand sides are evaluated at
p = r(u, v). Since k1 , k2 are the roots of λ2 − 2Hλ + K = 0, we have
p
k1 , k2 = H ± H 2 − K.
It is always nice to have a matrix that stores so much information (and perhaps surprising that all
of this information is therefore encoded in ν∗ ).
Part of The Fundamental Theorem of Surface Theory (Bonnet 1867) says that the functions
gij , `ij determine the immersion up to proper Euclidean motions. However, an arbitrary choice of
gij , `ij will not necessarily define a surface; we would like to determine the conditions under which
they do. Before exploring this point further, we should remind the reader of some terminology and
introduce some notation.
The Christoffel symbols Γkij of a Riemannian manifold M are the structure constants of the
Levi-Civita connection ∇ : X(M ) × X(M ) → X(M ), defined by
∇∂i ∂j =
n
X
Γkij ∂k
k=1
∂
where χ = (x1 , x2 , . . . , xn ) is a coordinate system on M and ∂i = ∂x
i are the corresponding constant
vectors fields. The Christoffel symbols can be written in terms of the metric as follows
n
o
X
g kl n
(gj` )i + (gi` )j − (gij )` .
2
Γkij =
(1)
`=1
The Riemannian curvature tensor R : X(M ) × X(M ) × X(M ) → X(M ) of a Riemannian
manifold is defined by
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z
for vector fields X, Y, Z on M . The curvature tensor therefore measures the non-commutativity
of the Levi-Civita connection. Again, in a coordinate system it is natural to define the structure
i
and Rijk` by
constants of this map Rjk`
R(∂k , ∂` )∂j =
n
X
i
Rjk`
∂i ,
and hR(∂k , ∂` )∂j , ∂i i = Rijk`
i=1
Now we can get back to the question of what the necessarily constraints are on the functions
gij , `ij . Differentiating the ri , using the definition hri , rj i = gij , and playing with identities involving
the Christoffel symbols leads to
rik =
2
X
Γhik rh + `ik N,
The Gauss Equations.
h=1
Differentiating N and defining `hi =
Ni = −
2
X
P2
j=1 g
`hi rh ,
hj `
ij
leads to
The Weingarten Equations.
h=1
The Gauss and Weingarten equations constitute an analogue of the Serret-Frenet formulas for a
curve – the derivatives of r1 , r2 , N have been expressed in terms of themselves. Hence, in order to
produce an immersion r with given gij and `ij we need to solve the Gauss and Weingarten equations.
Dynamics at the Horsetooth
4
Vol. 1, 2009
PDEs and surfaces in Lie algebras
Luke Bayens
However, these are partial differential equations (15 equations in the 9 component functions rij , N j ),
and these equations have solutions only if certain compatibility conditions are satisfied. Setting
the mixed partial derivatives rijk equal and using the linear independence of r1 , r2 , N gives
Γρik
j
2 X
− Γρij +
Γhik Γρhj − Γhij Γρhk = `ik `ρj − `ij `ρk
k
and
(`ik )j − (`ij )k +
Now the left hand side of (2) is
(2)
h=1
equal3
to
2
X
h=1
ρ
Rkji ,
Rhkji =
Γhik `hj −
2
X
Γhij `hk = 0.
(3)
h=1
and using
2
X
ρ
ghρ Rkji
ρ=1
we get
Rhkji = `hj `ik − `hk `ij .
Using r as the inverse of a coordinate system, we have the special case
R1212 = `11 `22 − `12 `12 = eg − f 2 .
This is equivalent to Gauss’ Theorema Egregium, since it says that the intrinsically defined
curvature K is given by
K=
eg − f 2
hR(r1 , r2 )r2 , r1 i
=
,
hr1 , r1 ihr2 , r2 i − hr1 , r2 i2
EG − F 2
where the right hand side is the extrinsically defined Gaussian curvature for a surface embedded
in R3 . Notice that we can use (2) to write K as follows:
R1212
EG − F 2
1
1
2
E R212
+ F R212
=
2
EG − F
n
1
1
1
1 1
1 1
2 1
2 1
=
E
Γ
−
Γ
+
Γ
Γ
−
Γ
Γ
+
Γ
Γ
−
Γ
Γ
22
21
22
11
21
12
22
21
21
22
1
2
EG − F 2
o
+ F Γ222 1 − Γ221 2 + Γ122 Γ211 − Γ112 − Γ212 .
K=
(4)
This expression for K shows that the Gaussian curvature depends only on the coefficients E, F, G
for the first fundamental form and their first derivatives. It is therefore a truly intrinsic property
of the Riemannian structure.
Let us now look at (3). Taking j = 1, k = 2 and i = 1 or 2 we get

2
2
X
X


h

e
−
f
=
Γ
`
−
Γh12 `h1

1
11 h2

 2
h=1
h=1
The Mainardi-Codazzi Equations.
2
2

X
X



Γh21 `h2 −
Γh22 `h2

 f2 − g1 =
h=1
3
h=1
This follows from a direct but slightly unpleasant computation.
Dynamics at the Horsetooth
5
Vol. 1, 2009
PDEs and surfaces in Lie algebras
Luke Bayens
It turns out that setting Nij = Nji gives us nothing new – these equations reduce to the MainardiCodazzi equations. The second part of The Fundamental Theorem of Surface Theory says answers
the question about sufficient conditions for gij and `ij to define a surface. Suppose we are given
a convex open set U ⊂ R2 containing (0, 0), and functions gij = gji and `ij = `ji on U with
(gij ) positive definite. Suppose further that gij and `ij satisfy both Gauss’ equations and the
Mainardi-Codazzi equations. Then there is an immersion r : U → R3 satisfying
gij = hri , rj i and `ij = h−Ni , rj i = hN, rji i,
for
r1 × r2
N=q
.
g11 g22 − (g12 )2
We can write the Gauss and Weingarten equations in matrix form as follows. Defining
  

r1
r1
Ψ = r2  =  r2  ,
r1 ×r2
N
EG−F 2

 1
  1
Γ11
Γ211
e
Γ11
Γ211 `11
Γ212
f ,
Γ212 `12  =  Γ112
S =  Γ112
eF
−f E
f
F
−eG
1
2
−`1
−`1
0
0
eG−F 2
eG−F 2
 1
  1

2
2
Γ12
Γ12
f
Γ12 Γ12 `12
Γ222
g ,
T =  Γ122 Γ222 `22  =  Γ122
gF −f G
f F −gE
1
2
−`1 −`1 0
0
eG−F 2
eG−F 2
the Gauss and Weingarten equations can be written concisely as
(
Ψ1 = SΨ
Ψ2 = T Ψ.
Applying the compatibility condition Ψ12 = Ψ21 , the Mainardi-Codazzi equations and the
Theorema Egregium can be written in the very appealing form
T1 − S2 + [T, S] = 0.
3
The sine-Gordon equation
Let us now consider immersing a particular surface of revolution in R3 . These are the surfaces
obtained by starting with a curve (the profile curve) defined in the right half of the (x, z)-plane,
and revolving it around the z-axis. Let us require that if the curve intersects the z-axis, it does so in
a right angle. If we parametrize the curve in the (x, z)-plane by c(s) = (g(s), h(s)), then the surface
of revolution is immersed in R3 by r(s, t) = (g(s) cos(t), g(s) sin(t), h(s)). If the parametrization
of c is canonical (that is, |c0 |2 = (g 0 )2 + (h0 )2 = 1), then we can use the equations in the previous
00
section to compute K = − gg . Our goal is to find a surface of revolution with constant negative
curvature.
Suppose K = − ρ12 . Then we need to solve ρ2 g 00 − g = 0, which has the general solution
√
g(s) = aes/
ρ
√
+ be−s/
ρ.
If a = 1 and b = 0 then we can take
Z sp
√
√
s/ ρ
g(s) = e
, h(s) = ±
1 − e2t/ ρ dt.
0
Dynamics at the Horsetooth
6
Vol. 1, 2009
PDEs and surfaces in Lie algebras
Luke Bayens
√
Clearly e2s/ ρ ≤ 1, and therefore g(s) ≤ 1. The resulting surface is a pseudosphere. Its profile
curve is a tractrix, and has the property that the length between a point P on the curve and the
intersection of the tangent line at P and the z-axis is constant. The upper tractrix is the graph of
the function
Z √ρ log(x) p
√
√ p
f (x) =
1 − e2t/ ρ dt = ρ
1 − x2 − cosh−1 (1/x) .
0
We are now going to choose a different parametrization of the pseudosphere. An asymptotic
curve c on a manifold M is a curve such that c0 always points along an asymptotic direction, that
is, if c00 lies in Tc(t) M for all t. A pseudosphere can be parametrized by asymptotic curves, and
so let r = (u, v) denote the inverse of such a coordinate system parametrized by arclength. Let
ω denote the angle between the asymptotic curves. It follows that the fundamental forms with
respect to these local coordinates are
I = du ⊗ du + cos(ω) du ⊗ du + dv ⊗ dv
1
II = sin(ω) du ⊗ dv,
ρ
where we have used the Theorema Egregium to deduce the du ⊗ dv term of II. We can now use
the fundamental forms to compute the Christoffel symbols using (1):
Γ111 = cot(ω) ω1
Γ211 = − csc(ω) ω1
Γ122 = − csc(ω) ω2
Γ222 = cot(ω) ω2 ,
and Γkij = 0 when i 6= j. Substituting these equations into (4), and using the fact that K = − ρ12
we find that ω satisfies
ω12 =
1
sin(ω),
ρ2
The sine-Gordon equation.
Using our matrix notation for the system of equations satisfied by this embedding we get

ω1 cos(ω) −ω1 csc(ω)
0
0
Ψ1 = 
1
1
− ρ csc(ω)
ρ cot(ω)

0
0

Ψ2 = −ω2 csc(ω) ω2 cot(ω)
− ρ1 csc(ω) ρ1 cot(ω)

0
1

ρ sin(ω) Ψ
0

1
ρ sin(ω)
0 Ψ
0
and the compatibility condition T1 − S2 + [S, T ] = 0 for these equations is the sine-Gordon equation
ω12 = ρ12 sin(ω).
This representation is certainly much more pleasing than the earlier expressions we had for the
Theorema Egregium, Gauss, Weingarten and Mainardi-Codazzi equations; however we can still do
better. One issue is that our frame (r1 , r2 , N ) is not orthonormal. This is easily fixed by defining
A = r1 ,
B = −r1 × N,
C = N.
Expressing the above equations in this basis we have
Ψ1 = SΨ,
Dynamics at the Horsetooth
Ψ2 = T Ψ
7
Vol. 1, 2009
PDEs and surfaces in Lie algebras
Luke Bayens
where
 
A

Ψ = B
C


0 −ω1 0
1
0
S = ω1
ρ
0 − ρ1 0


1
0
0
ρ sin(ω)

0
0
− ρ1 cos(ω)
T =

− ρ1 sin(ω) ρ1 cos(ω)
0
Notice that Ψ ∈ SO(3), and S, T ∈ so(3), the corresponding Lie algebra of skew-symmetric, traceless
matrices. We would like to reduce the dimension of these matrices while preserving the essential
structure – this is achieved by an isomorphism between so(3) and su(2). Let us take the basis






0 −1 0
0 0 1
0 0 0
L1 = 0 0 −1 , L2 =  0 0 0 , L3 = 1 0 0
0 0 0
−1 0 0
0 1 0
of so(3). In terms of this basis,
1
S = ω1 L3 − L1 ,
ρ
1
1
T = cos(ω)L1 + sin(ω)L2 .
ρ
ρ
Now take the basis
1
e1 =
2i
0 1
,
1 0
1
e2 =
2i
0 −i
,
i 0
1
e3 =
2i
1 0
0 −1
of su(2). The map Li ↔ ei is a Lie algebra isomorphism between so(3) and su(2). Under this map
S, T are sent to U, V respectively, where
!
1
i −ω1 ρ1
U = ω1 e3 − e1 =
1
ω1
ρ
2
ρ
1
1
i
0 e−iω
V = cos(ω) e1 + sin(ω) e2 = −
0
ρ
ρ
2ρ eiω
The sine-Gordon equation is now the compatibility condition P1 − Q2 + [P, Q] = 0 of the linear
system
Φ1 = U Φ
(5)
Φ2 = V Φ.
4
Surfaces in Lie groups and Lie algebras
We will now generalize the approach in the last section. For any pseudospherical surface immersed
in R3 we constructed su(2) valued functions U, V and an SU(2) valued function Φ such that (5)
holds. We can find pseudospherical surfaces by solving the compatibility conditions for (5), which is
Dynamics at the Horsetooth
8
Vol. 1, 2009
PDEs and surfaces in Lie algebras
Luke Bayens
equivalent to solving the sine-Gordon equation. Given an immersion r : U → R3 defining a surface
in R3 , we can consider r as a map into su(2) given by r(u, v) = r1 (u, v)e1 + r2 (u, v)e2 + r3 (u, v)e3 ,
where ei are the basis for su(2) given in the previous section and ri are the component functions
of r. Thus, for every immersed pseudospherical surface we have a surface in su(2), two associated
maps U, V into su(2), and a map Φ into SU(2). The idea is to generalize this situation to an
arbitrary surface in a Lie algebra, associating to it the appropriate functions U, V, Φ expressing the
Mainardi-Codazzi equations and the Theorema Egregium.
Let G be a Lie group with corresponding Lie algebra g with an invariant non-degenerate
symmetric bilinear form h , i. For the moment let us suppose that we have G is an n dimensional
matrix Lie group and g its associated n dimensional matrix Lie algebra. A surface in g is given by
an immersion r : U → g, with U ⊂ R2 open. The first fundamental form is defined by
I = hr1 , r1 i du ⊗ du + hr1 , r2 i du ⊗ dv + hr2 , r1 i dv ⊗ du + hr2 , r2 i dv ⊗ dv
For k = 1, 2, . . . , n − 2, we can define N (k) ∈ g by
hN (k) , N (k) i = 1,
hr1 , N (k) i = hr2 , N (k) i = 0,
so that the second fundamental forms of r are defined by
II = hr11 , N (k) i du ⊗ du + hr12 , N (k) i du ⊗ dv + hr21 , N (k) i dv ⊗ du + hr22 , N (k) i dv ⊗ dv.
We can also define the notion of a surface in a Lie group. The functions U, V : Ω → g, Ω ⊂ R2
open, define a map Φ : Ω → G provided that the compatibility condition U1 − V2 + [U, V ] = 0 holds
(that is, if the equations given by (5) have a unique solution). We call Φ a surface in G.
The relationship between surfaces in g are surfaces in G has been studied in [4, 2, 1] and is
encapsulated in the following proposition.
Theorem ([4]). Let Ω ⊂ R2 , Ω open, and let U, V : Ω → g define a surface in G via (5). Let
A, B : Ω → g. Then the equations
r1 = Φ−1 AΦ,
r2 = Φ−1 BΦ
define a surface r in g if and only if A and B satisfy
A + [A, Φ2 Φ−1 ] = B1 + [B, Φ1 Φ−1 ].
This is simply a restatement of the compatibility conditions for the linear system of equations.
Since the form on g is invariant under the adjoint representation, we can compute the first and
second fundamental forms of r as functions of A, B, U, V .
We have only touched the surface of the relationship between Lie theory and integrable
equations, differential geometry and their associated surfaces. Indeed, we are going to end this
paper just as things are starting to get interesting. There is a limit to how much can be covered in
only a handful of pages, and sadly, we have reached this limit.
Dynamics at the Horsetooth
9
Vol. 1, 2009
PDEs and surfaces in Lie algebras
Luke Bayens
References
[1] H. Abbaspour and M. Moskowitz, Basic Lie theory, World Scientific Publishing Co. Pte. Ltd.,
Hackensack, NJ, 2007.
[2] Ö. Ceyhan, A. S. Fokas, and M. Gürses, Deformations of surfaces associated with integrable
Gauss-Mainardi-Codazzi equations, J. Math. Phys. 41 (2000), no. 4, 2251–2270.
[3] M.P. do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser
Boston Inc., Boston, MA, 1992, Translated from the second Portuguese edition by Francis
Flaherty.
[4] A. S. Fokas and I. M. Gelfand, Surfaces on Lie groups, on Lie algebras, and their integrability,
Comm. Math. Phys. 177 (1996), no. 1, 203–220, With an appendix by Juan Carlos Alvarez
Paiva.
[5] C.F. Gauss, General investigations of curved surfaces, Translated from the Latin and German
by Adam Hiltebeitel and James Moreh ead, Raven Press, Hewlett, N.Y., 1965.
[6] J.M. Lee, Riemannian manifolds, Graduate Texts in Mathematics, vol. 176, Springer-Verlag,
New York, 1997, An introduction to curvature.
[7] M. Spivak, A comprehensive introduction to differential geometry. Vol. I, second ed., Publish
or Perish Inc., Wilmington, Del., 1979.
Dynamics at the Horsetooth
10
Vol. 1, 2009