The geometry of a class of partial differential equations and surfaces in Lie algebras

Transcription

1 Dynamics at the Horsetooth Volume, 29. The geometry of a class of partial differential equations and surfaces in Lie algebras Department of Mathematics Colorado State University Report submitted to Prof. P. Shipman for Math 54, Fall 29 bayens@math.colostate.edu Abstract. We show how an immersion of a surface in R 3 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 Introduction The most important work in the history of differential geometry is Gauss 828 paper Disquisitiones generales circa superficies curvas [5]. In this work Gauss describes properties of surfaces embedded in R 3, 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 R 3 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 R 3 and surfaces in Lie groups and Lie algebras.

2 2 Elements of the theory of surfaces in R 3 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 R 3, with i : M R 3 the inclusion map. The first fundamental form I is I = i,, where, is the usual Riemannian metric on R 3. (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 R 3 for some open U R 2, then E(r(u, v)) = r (u, v), r (u, v) F (r(u, v)) = r (u, v), r 2 (u, v) G(r(u, v)) = r 2 (u, v), r 2 (u, v), where r i denotes the partial derivative of r with respect to the ith variable. Here we see our first abuse of notation when we write r i (u, v) we are thinking of r i (u, v) T r(u,v) M which we are identifying with T r(u,v) R 3. We should really write r i (u, v) r(u,v) T r(u,v) M T r(u,v) R 3 or something similar, but for obvious reasons we will not. Since E = r, r r, 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 R 3 is an immersion, the first fundamental form I r of r is defined to be the tensor r, on M. In particular, when r : U R 3 for open U R 2 we have a form I r on U defined by I r (u, v)(x, X 2 ) = r X, r X 2, for X, X 2 T (u,v) R 2. We can then define functions E, F, G on U by E = r, r, F = r, r 2, G = r 2, r 2. These are nothing more than the components of I r = r, with respect to the standard coordinate system (u, v) on R 2. We introduce the subscript notation g ij = r i, r j, so that g = E, g 2 = g 2 = F, g 22 = G, and the superscript notation g ij such that k g ij g kj = δ j i, that is, (g ij) = (g ij ). Since r, is positive definite, it follows that det(g ij ) = EG F 2 > and r r 2 = EG F 2. For every p M, we can consider the tangent space T p M as a subspace of T p R 3 by identifying T p M with i (T p M) T i(p) R 3 = T p R 3. In the vector space T p R 3, with inner product, p, the subspace T p M has an orthogonal complement (T p M) T p R 3, and we can use the decomposition T p R 3 = T p M (T p M) to write any v T p R 3 as v = v T + v N for v T T p M (the tangential projection of v) and v N (T p 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 ν, ν =, ν(q) (T p M), and (X, X 2 ) T p M is positively oriented if and only if (X, X 2, ν(p)) is positively oriented in R 3. We will usually regard ν : M S 2 R 3 and write this normal vector field in the very abusive form p ν(p) p T p M. 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., 29

3 If instead we are dealing with an immersion r : M R 3, 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 N(q) r(q) T r(q) R 3. Here N is a function N : M S 2 R 3. We will adopt the convention of using ν when considering embedded submanifolds M R 3 and N when considering immersions r : M R 3. If W M is an open set on which r is an embedding, then a unit normal field ν on r(w ) R 3 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)(X, X 2 ) = ν X, X 2, for X, X 2 T p M. This is once again an abuse of notation; we are identifying 2 T ν(p) R 3 T p R 3 and thinking of ν : T p M T p M so that the inner product makes sense. } Notice that the matrix of ν : T p M T p M with respect to the ordered basis {(r ) p, (r 2 ) p is (g ij ) (l ij ). Similarly, we define the second fundamental form II r of r to be the tensor on M defined by II r (q)(x, X 2 ) = N X, r X 2, for X, X 2 T q M, with the same identifications. Equivalently, we have II r = r II. In particular, consider an immersion r : U R 3, for U R 2 open. We can choose N explicitly to be N = r r 2 r r 2 = r r 2. EG F 2 Then II r (u, v)(x, X 2 ) = N X, r X 2, for X, X 2 T (u,v) R 2. We can define the functions e, f, g on U by e = N, r = N, r, f = N, r 2 = N, r 2, g = N 2, r 2 = N, r 22. Thus e, f, g are simply the components of II r with respect to the standard coordinate system (u, v) on R 2. Again, it is sometimes convenient to use the subscript notation l ij = N i, r j = N, r ij. At this point we can see } some use for all this notation. The matrix M of ν : T p M T p M with respect to {(r ) p, (r 2 ) p is M = (g ij ) (l ij ) ( ) ( ) G F e f = EG F 2 F E f g with r i, g ij, l ij are evaluated at (u, v), where p = r(u, v). The principle curvatures k, k 2 are the eigenvalues of M, the Gaussian curvature K is k k 2, and the mean curvature H is (k + k 2 )/2. So K and H are the determinant and half the trace, respectively, of M. Thus, K = H = ef g2 EG F 2 Eg 2F f + Ge 2(EG F 2, ) 2 Here we are using the Levi-Civita connection of the standard Riemannian metric on R 3. Dynamics at the Horsetooth 3 Vol., 29

4 where the left hand sides must be evaluated at (u, v) when the right hand sides are evaluated at p = r(u, v). Since k, k 2 are the roots of λ 2 2Hλ + K =, we have k, k 2 = 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 867) says that the functions g ij, l ij determine the immersion up to proper Euclidean motions. However, an arbitrary choice of g ij, l 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 Γ k ij 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 k= Γ k ij k where χ = (x, x 2,..., x n ) is a coordinate system on M and i = are the corresponding constant x i vectors fields. The Christoffel symbols can be written in terms of the metric as follows Γ k ij = n l= g kl 2 {(g jl ) i + (g il ) j (g ij ) l }. () 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 constants of this map R i jkl and R ijkl by R( k, l ) j = n Rjkl i i, and R( k, l ) j, i = R ijkl i= Now we can get back to the question of what the necessarily constraints are on the functions g ij, l ij. Differentiating the r i, using the definition r i, r j = g ij, and playing with identities involving the Christoffel symbols leads to r ik = Γ h ik r h + l ik N, h= The Gauss Equations. Differentiating N and defining l h i = 2 j= ghj l ij leads to N i = l h i r h, h= The Weingarten Equations. The Gauss and Weingarten equations constitute an analogue of the Serret-Frenet formulas for a curve the derivatives of r, r 2, N have been expressed in terms of themselves. Hence, in order to produce an immersion r with given g ij and l ij we need to solve the Gauss and Weingarten equations. Dynamics at the Horsetooth 4 Vol., 29

5 However, these are partial differential equations (5 equations in the 9 component functions r j i, N j ), and these equations have solutions only if certain compatibility conditions are satisfied. Setting the mixed partial derivatives r ijk equal and using the linear independence of r, r 2, N gives and ( ( Γ ρ ik )j Γ ρ ij )k + h= (l ik ) j (l ij ) k + ( ) Γ h ik Γρ hj Γh ijγ ρ hk = l ik l ρ j l ijl ρ k (2) Γ h ik l hj h= Now the left hand side of (2) is equal 3 to R ρ kji, and using we get R hkji = Γ h ijl hk =. (3) h= g hρ R ρ kji ρ= R hkji = l hj l ik l hk l ij. Using r as the inverse of a coordinate system, we have the special case R 22 = l l 22 l 2 l 2 = eg f 2. This is equivalent to Gauss Theorema Egregium, since it says that the intrinsically defined curvature K is given by K = R(r, r 2 )r 2, r r, r r 2, r 2 r, r 2 2 = eg f 2 EG F 2, where the right hand side is the extrinsically defined Gaussian curvature for a surface embedded in R 3. Notice that we can use (2) to write K as follows: K = R 22 EG F 2 { = E R EG F F R22 2 } { = EG F 2 E ( ( Γ ) 22 ( Γ ) Γ 22Γ Γ 2Γ 2 + Γ 2 22Γ 2 Γ 2 2Γ ) 22 + F ( ( Γ 2 ) 22 ( Γ 2 ) Γ 22Γ 2 Γ 2 Γ 2 2) }. (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 =, k = 2 and i = or 2 we get e 2 f = Γ h l h2 Γ h 2l h h= h= The Mainardi-Codazzi Equations. f 2 g = Γ h 2l h2 Γ h 22l h2 h= h= 3 This follows from a direct but slightly unpleasant computation. Dynamics at the Horsetooth 5 Vol., 29

6 It turns out that setting N ij = N ji gives us nothing new these equations reduce to the Mainardi- Codazzi equations. The second part of The Fundamental Theorem of Surface Theory says answers the question about sufficient conditions for g ij and l ij to define a surface. Suppose we are given a convex open set U R 2 containing (, ), and functions g ij = g ji and l ij = l ji on U with (g ij ) positive definite. Suppose further that g ij and l ij satisfy both Gauss equations and the Mainardi-Codazzi equations. Then there is an immersion r : U R 3 satisfying g ij = r i, r j and l ij = N i, r j = N, r ji, for N = r r 2 g g 22 (g 2 ) 2. We can write the Gauss and Weingarten equations in matrix form as follows. Defining r r Ψ = r 2 = r 2, r N r 2 EG F 2 Γ Γ 2 l Γ S = Γ 2 Γ 2 Γ 2 e 2 l 2 = Γ l l 2 2 Γ 2 2 f, ff eg ef fe eg F 2 eg F 2 Γ 2 Γ 2 2 l 2 Γ T = Γ 22 Γ 2 2 Γ 2 2 f 22 l 22 = Γ l l 2 22 Γ 2 22 g, gf fg eg F 2 the Gauss and Weingarten equations can be written concisely as { Ψ = SΨ Ψ 2 = T Ψ. ff ge eg F 2 Applying the compatibility condition Ψ 2 = Ψ 2, the Mainardi-Codazzi equations and the Theorema Egregium can be written in the very appealing form 3 The sine-gordon equation T S 2 + [T, S] =. Let us now consider immersing a particular surface of revolution in R 3. 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 R 3 by r(s, t) = (g(s) cos(t), g(s) sin(t), h(s)). If the parametrization of c is canonical (that is, c 2 = (g ) 2 + (h ) 2 = ), then we can use the equations in the previous section to compute K = g g. Our goal is to find a surface of revolution with constant negative curvature. Suppose K =. Then we need to solve ρ 2 g g =, which has the general solution ρ 2 g(s) = ae s/ ρ + be s/ ρ. If a = and b = then we can take g(s) = e s/ ρ, h(s) = ± s e 2t/ ρ dt. Dynamics at the Horsetooth 6 Vol., 29

7 Clearly e 2s/ ρ, and therefore g(s). 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 f(x) = ρ log(x) e 2t/ ρ dt = ( ) ρ x 2 cosh (/x). 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 c always points along an asymptotic direction, that is, if c lies in T c(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 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 (): Γ = cot(ω) ω Γ 2 = csc(ω) ω Γ 22 = csc(ω) ω 2 Γ 2 22 = cot(ω) ω 2, and Γ k ij = when i j. Substituting these equations into (4), and using the fact that K = ρ 2 we find that ω satisfies ω 2 = ρ 2 sin(ω), The sine-gordon equation. Using our matrix notation for the system of equations satisfied by this embedding we get ω cos(ω) ω csc(ω) Ψ = ρ sin(ω) Ψ ρ cot(ω) ρ csc(ω) ρ sin(ω) Ψ 2 = ω 2 csc(ω) ω 2 cot(ω) Ψ ρ csc(ω) ρ cot(ω) and the compatibility condition T S 2 +[S, T ] = for these equations is the sine-gordon equation ω 2 = ρ 2 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 (r, r 2, N) is not orthonormal. This is easily fixed by defining A = r, B = r N, C = N. Expressing the above equations in this basis we have Ψ = SΨ, Ψ 2 = T Ψ Dynamics at the Horsetooth 7 Vol., 29

8 where A Ψ = B C ω S = ω ρ ρ ρ T = ρ cos(ω) ρ sin(ω) ρ cos(ω) 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 of so(3). In terms of this basis, L =, L 2 =, L 3 = S = ω L 3 ρ L, T = ρ cos(ω)l + ρ sin(ω)l 2. Now take the basis e = 2i ( ), e 2 = 2i ( ) i, e i 3 = ( ) 2i of su(2). The map L i e i is a Lie algebra isomorphism between so(3) and su(2). Under this map S, T are sent to U, V respectively, where ( ) U = ω e 3 ρ e = i ω ρ 2 ρ ω V = ρ cos(ω) e + ρ sin(ω) e 2 = i ( ) e iω 2ρ e iω The sine-gordon equation is now the compatibility condition P Q 2 + [P, Q] = of the linear system Φ = UΦ (5) Φ 2 = 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 R 3 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., 29

9 equivalent to solving the sine-gordon equation. Given an immersion r : U R 3 defining a surface in R 3, we can consider r as a map into su(2) given by r(u, v) = r (u, v)e + r 2 (u, v)e 2 + r 3 (u, v)e 3, where e i are the basis for su(2) given in the previous section and r i 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,. 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 R 2 open. The first fundamental form is defined by I = r, r du du + r, r 2 du dv + r 2, r dv du + r 2, r 2 dv dv For k =, 2,..., n 2, we can define N (k) g by N (k), N (k) =, r, N (k) = r 2, N (k) =, so that the second fundamental forms of r are defined by II = r, N (k) du du + r 2, N (k) du dv + r 2, N (k) dv du + r 22, N (k) dv dv. We can also define the notion of a surface in a Lie group. The functions U, V : Ω g, Ω R 2 open, define a map Φ : Ω G provided that the compatibility condition U V 2 + [U, V ] = 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, ] and is encapsulated in the following proposition. Theorem ([4]). Let Ω R 2, Ω open, and let U, V : Ω g define a surface in G via (5). Let A, B : Ω g. Then the equations r = Φ AΦ, r 2 = Φ BΦ define a surface r in g if and only if A and B satisfy A + [A, Φ 2 Φ ] = B + [B, Φ Φ ]. 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., 29

10 References [] H. Abbaspour and M. Moskowitz, Basic Lie theory, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 27. [2] Ö. Ceyhan, A. S. Fokas, and M. Gürses, Deformations of surfaces associated with integrable Gauss-Mainardi-Codazzi equations, J. Math. Phys. 4 (2), no. 4, [3] M.P. do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 992, 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. 77 (996), no., 23 22, 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., 965. [6] J.M. Lee, Riemannian manifolds, Graduate Texts in Mathematics, vol. 76, Springer-Verlag, New York, 997, An introduction to curvature. [7] M. Spivak, A comprehensive introduction to differential geometry. Vol. I, second ed., Publish or Perish Inc., Wilmington, Del., 979. Dynamics at the Horsetooth Vol., 29