Polynomial conserved quantities of Lie applicable surfaces
Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and -isothermic surfaces of Laguerre geometry. In this setting one can see that the well known transformations available for these surfaces are induced by the transformations of the underlying Lie applicable surfaces. We also consider linear Weingarten surfaces in this setting and develop a new Bäcklund-type transformation for these surfaces.
given in terms of curvature line coordinates , where is a function of , is a function of , , and denote the usual coefficients of the first fundamental form and and denote the principal curvatures. In the case that , one calls these surfaces -surfaces and if we call them -surfaces. Together, - and -surfaces form the applicable surfaces of Lie sphere geometry (see ). By using the hexaspherical coordinate model of Lie  it is shown in  that these surfaces are the deformable surfaces of Lie sphere geometry. This gives rise to a gauge theoretic approach for these surfaces which is developed in . That is, the definition of Lie applicable surfaces is equated to the existence of a certain 1-parameter family of flat connections. This approach lends itself well to the transformation theory of these surfaces. The gauge theoretic approach is explored further in , for which this paper is meant as a sequel.
In [11, 37] a gauge theoretic approach for isothermic surfaces in Möbius geometry is developed. By considering polynomial conserved quantities of the arising 1-parameter family of flat connections, one can characterise familiar subclasses of surfaces in certain space forms. For example, constant mean curvature surfaces in space forms are characterised by the existence of linear conserved quantities [4, 11]. By then applying the transformation theory of the underlying isothermic surface, one obtains transformations for these subclasses. In this paper we apply this framework to Lie applicable surfaces. For example, we show that isothermic surfaces, Guichard surfaces and -isothermic surfaces are -surfaces admitting a linear conserved quantity. This is particularly beneficial to the study of the transformations of these surfaces. For example, we will show that the Eisenhart transformation for Guichard surfaces (see ), which was given a conformally invariant treatment in , is induced by the Darboux transformation of the underlying -surface. One can also show (see ) that the special -surfaces of  can be characterised as -surfaces admitting quadratic conserved quantities, however we will not explore this further in this paper.
In [8, 9] linear Weingarten surfaces in space forms are characterised as Lie applicable surfaces whose isothermic sphere congruences take values in certain sphere complexes. In this paper we shall review this theory from the viewpoint of polynomial conserved quantities. We shall see that non-tubular linear Weingarten surfaces in space forms are -surfaces admitting a 2-dimensional vector space of linear conserved quantities, whereas tubular linear Weingarten surfaces are -surfaces admitting a constant conserved quantity. By using this approach we obtain a new Bäcklund-type transformation for linear Weingarten surfaces.
Acknowledgements. We would like to thank G. Szewieczek for reading through this paper and providing many useful comments. This work has been partially supported by the Austrian Science Fund (FWF) through the research project P28427-N35 “Non-rigidity and Symmetry breaking” as well as by FWF and the Japan Society for the Promotion of Science (JSPS) through the FWF/JSPS Joint Project grant I1671-N26 “Transformations and Singularities”. The fourth author was also supported by the two JSPS grants Grant-in-Aid for Scientific Research (C) 15K04845 and (S) 24224001 (PI: M.-H. Saito).
Given a vector space and a manifold , we shall denote by the trivial bundle . Given a vector subbundle of , we define the derived bundle of , denoted , to be the subset of consisting of the images of sections of and derivatives of sections of with respect to the trivial connection on . In this paper, most of the derived bundles that appear will be vector subbundles of the trivial bundle, but in general this is not always the case as, for example, the rank of the derived bundle may not be constant over .
Throughout this paper we shall be considering the pseudo-Euclidean space , i.e., a 6-dimensional vector space equipped with a non-degenerate symmetric bilinear form of signature . Let denote the lightcone of . The orthogonal group acts transitively on . The lie algebra of is well known to be isomorphic to the exterior algebra via the identification
for . We shall frequently use this fact throughout this paper.
By we shall denote the symmetric product on that for gives , i.e.,
Given a manifold , we define the following product of two vector-valued 1-forms :
for . Hence, is a -form taking values in . Notice that .
Recall that we also have the following product for two -valued 1-forms :
2.1. Legendre maps
Let denote the Grassmannian of isotropic 2-dimensional subspaces of . Suppose that is a 2-dimensional manifold and let be a smooth map. By viewing as a 2-dimensional subbundle of the trivial bundle , we may define a tensor, analogous to the solder form defined in [5, 10],
In accordance with [12, Theorem 4.3] we have the following definition:
is a Legendre map if satisfies the contact condition, , and the immersion condition, .
The contact and immersion conditions together imply that (see ).
Note that is a rank subbundle of that inherits a positive definite metric from .
Let . Then a 1-dimensional subspace is a curvature sphere of at if there exists a non-zero subspace such that . We call the maximal such the curvature space of .
It was shown in  that at each point there is either one or two curvature spheres. We say that is an umbilic point of if there is exactly one curvature sphere at and in that case .
Suppose that such that . Then is totally umbilic.
If then . Therefore implies that is a curvature sphere congruence of with curvature subbundle . By the immersion condition of this is the only curvature sphere congruence of and thus is totally umbilic. ∎
Away from umbilic points we have that the curvature spheres form two rank 1 subbundles with respective curvature subbundles and . We then have that and . A conformal structure is induced on as the set of all indefinite metrics whose null lines are and .
2.2. Symmetry breaking
In  a modern account is given of how one breaks symmetry from Lie geometry to space form geometry and how is a double cover for the group of Lie sphere transformations. These are the transformations that map oriented spheres to oriented spheres and preserve the oriented contact of spheres. In this subsection we shall recall the process of symmetry breaking.
Firstly, we have the following technical result regarding projections of Legendre maps:
Suppose that is a Legendre map and . Then
if is timelike then never belongs to ,
if is spacelike then the set of points where is a closed set with empty interior,
if and only if , in which case is totally umbilic.
If is timelike then has signature and cannot contain the 2-dimensional lightlike subspace for each .
Suppose that is spacelike and that on some open subset , . Without loss of generality, assume that . Then this implies that and . Hence, , contradicting Remark 2.2.
Therefore, if then the only possibility left to consider is that is lightlike. Then since the maximal lightlike subspaces of are 2-dimensional, if and only if . By Lemma 2.4 this is the case only if is totally umbilic. ∎
We shall often refer to a non-zero vector as a sphere complex. As Lemma 2.5 shows, for a Legendre map , generically defines a rank 1 subbundle of .
2.2.1. Conformal geometry
Let such that is not lightlike. If is timelike then and defines a Riemannian conformal geometry. If is spacelike then and defines a Lorentzian conformal geometry. We consider elements of
to be points and refer to as a point sphere complex.
In the case that is timelike, is the conformal -sphere (see ).
The elements of give rise to spheres in the following way: suppose that . Now is a -plane and thus
is a -plane. The projective lightcone of is then diffeomorphic to and we thus identify with a sphere in .
Conversely, suppose that is a -plane. Then is a -plane in containing and we identify the two null lines of with the sphere defined by with opposite orientations.
Those Lie sphere transformations that fix the point sphere complex are the conformal transformations of .
As is standard in conformal geometry (see, for example, ), we may break symmetry further by choosing a vector . Then
is isometric to a space form with sectional curvature . If we assume that , then
can be identified (see ) with the space of hyperplanes (complete, totally geodesic hypersurfaces) in this space form.
Suppose that is a Legendre map. Then, by Lemma 2.5, on a dense open subset of , is a rank 1 subbundle of . Using the identification of with the skew-symmetric endomorphisms on , we have for any that and, since is skew-symmetric, . Hence,
Away from points where , we have that for any nowhere zero ,
are the projections of into and , respectively. We can then write .
We call the space form projection of and the tangent plane111Note that “plane” here means totally geodesic hypersurface in the space form . congruence of .
One can easily see that:
The space form projection of into exists at if and only if the kernel of the linear map
is trivial at .
Away from umbilic points, suppose that are curvature line coordinates for . Then by Rodrigues’ equations we have that
where and are the principal curvatures of . Therefore,
are curvature spheres of with respective curvature subbundles and .
2.2.2. Laguerre geometry
In this subsection we shall recall the correspondence given in  between Lie sphere geometry and Laguerre geometry. Let and define . Then with
defines an affine chart for . Choosing such that , we have that . We may then define the orthogonal projection
Then defines an isomorphism between and . We thus identify points in with points in . Now let . Then identifies with the projective lightcone of and thus , where is the lightcone of . Therefore, we identify with null directions in . We define to be the improper point of Laguerre geometry.
Under this correspondence, contact elements in are then identified with affine null lines in , i.e., for and
By choosing a point sphere complex with , we have that
One identifies points in with oriented spheres (including point spheres, but not oriented planes) in in the following way: a sphere centred at with signed radius is identified with the point
This is classically known as isotropy projection [3, 12]. We then have that null lines in correspond to pencils of spheres in in oriented contact with each other and isotropic planes in are identified with oriented planes in .
It was shown in  that the Lie sphere transformations that preserve the improper point are identified under this correspondence with the affine Laguerre transformations of , that is, the identity component of the group . In terms of transformations of , this group consists of the Lie sphere transformations that map oriented planes to oriented planes.
2.3. Lie applicable surfaces
Definition 2.10 ([34, Definition 3.1]).
We say that is a Lie applicable surface if there exists a closed such that and the quadratic differential defined by
is non-zero. Furthermore, if is non-degenerate (respectively, degenerate) on a dense open subset of we say that is an -surface (-surface).
Given a closed , we have for any that is a new closed 1-form taking values in . We then say that and are gauge equivalent and this yields an equivalence relation on closed 1-forms with values in . We call the equivalence class
the gauge orbit of . As shown in [34, Corollary 3.3], is well defined on gauge orbits, i.e., if and are gauge equivalent then , for their respective quadratic differentials.
Let us assume that is umbilic-free. Then there are two distinct curvature sphere congruences and with respective curvature subbundles and .
Proposition 2.11 ([34, Proposition 3.4]).
For an umbilic-free Legendre map , is closed if and only if satisfies the Maurer Cartan equation, i.e., . In this case, and .
One has a splitting of the trivial bundle called the Lie cyclide splitting:
for , , and . Since we may identify with , we have a splitting222This is the Cartan decomposition for the symmetric space of Dupin cyclides.
Therefore,we may split a closed 1-form into , accordingly. In [34, Definition 3.8] it is shown that there is a unique member of the gauge orbit of that satisfies . We call this unique member the middle potential and denote it by .
Assumption: for the rest of this paper we will make the assumption that the signature of the quadratic differential is constant over all of .
From Proposition 2.11, one can deduce that . Therefore, after possibly rescaling by and switching and , we may write
for unique (up to sign) lifts of the curvature sphere congruences and . The middle potential is then given by
where is the Hodge-star operator of the conformal structure for which the curvature directions on are null. One finds that is divergence-free with respect to , i.e., in terms of curvature line coordinates and , there exist functions of and of such that
When one projects to a space form, where the space form projection immerses, one finds that Demoulin’s equation
2.4. Transformations of Lie applicable surfaces
The transformation theory for Lie applicable surfaces was developed in  and was further explored in . In this section we shall review some of this theory. The richness of the transformation theory of Lie applicable surfaces follows from the following result:
Theorem 2.12 ([13, Lemma 4.2.6]).
Suppose that is closed and . Then is a 1-parameter family of flat metric connections.
Suppose now that , for some .
Lemma 2.13 ([13, Lemma 4.5.1]).
In the case that we are using the middle potential, , we shall refer to the 1-parameter family of connections as the middle pencil of connections, or for brevity, the middle pencil.
2.4.1. Calapso transforms
For each and gauge potential , since is a flat metric connection, there exists a local orthogonal trivialising gauge transformation , that is,
is called a Calapso transform of .
By Lemma 2.13, if is in the gauge orbit of , then the corresponding local orthogonal trivialising gauge transformations of are given by
Since , we have that the Calapso transforms are well defined on the gauge orbit.
In [34, Theorem 4.4] it is shown that is a closed 1-form taking values in . Furthermore, and . Thus we have the following theorem:
Calapso transforms are Lie applicable surfaces.
In fact, this 1-parameter family of Lie applicable surfaces arises because Lie applicable surfaces are the deformable surfaces of Lie sphere geometry (see ).
Proposition 2.16 ([34, Proposition 4.5]).
For any ,
are the local trivialising orthogonal gauge transformations of .
2.4.2. Darboux transforms
Fix and let be any gauge potential. Since is a flat connection, it has many parallel sections. Suppose that is a null rank 1 parallel subbundle of such that is nowhere orthogonal to the curvature sphere congruences of . Let and let .
is a Darboux transform of with parameter .
If , then by Lemma 2.13, we have that is a parallel subbundle of . However, and determine the same . Thus, Darboux transforms are invariant of choice of gauge potential in the gauge orbit of .
It was shown in [13, Theorem 4.3.7, Proposition 4.3.8] that is a Lie applicable surface, and is a Darboux transform of with parameter , i.e., for any gauge potential , there exists a parallel subbundle of . Thus:
Darboux transforms of Lie applicable surfaces are Lie applicable surfaces.
In the case that and are umbilic-free we have the following result regarding the middle pencils of the two surfaces:
Proposition 2.19 ([34, Proposition 4.17, Theorem 4.19]).
Suppose that and are umbilic-free Darboux transforms of each other with parameter . Then
where and are the parallel subbundles of and , respectively, implementing these Darboux transforms.
From  we also have the following proposition:
Proposition 2.20 ([34, Proposition 4.14]).
Suppose that is a Darboux transform of with parameter and let be any rank 2 subbundle of with . Then there exist gauge potentials and such that is a parallel subbundle of and is a parallel subbundle of .
In particular, this proposition shows that given any subbundle , one may choose a gauge potential such that is a parallel subbundle of .
A pertinent question is “how many Darboux transforms does a Lie applicable surface admit?” By using that
for every , one deduces the following lemma:
is a null rank 1 parallel subbundle of if and only if for some constant .
Now, since is 4-dimensional, admits a 4-parameter family of null rank 1 parallel subbundles. Since this holds for every , we obtain the following answer to our question:
2.5. Associate surfaces
Let and be a space form vector and point sphere complex with and , i.e.,
has sectional curvature and . Then we may choose a null vector such that . Thus and we have an isometry
We can use this to identify with a surface . Let denote the unit normal of . We then have that and the tangent plane congruence of is given by .
It was shown in [34, Section 5] that there exists a 1-parameter family of closed 1-forms in the gauge orbit of satisfying . We may then write
where and are Combescure transforms of , i.e., and have parallel curvature directions to , such that the principal curvatures of the surfaces satisfy
We call an associate surface of and an associate Gauss map of . In [34, Theorem 5.4] it was shown that an associate surface of an -surface is itself an -surface.
3. Polynomial conserved quantities
A non-zero polynomial is called a polynomial conserved quantity of if is a parallel section of for all .
The following lemma shows that the existence of polynomial conserved quantities is gauge invariant. Suppose that is in the gauge orbit of so that for . From Lemma 2.13 we immediately get the following result:
Suppose that is a polynomial conserved quantity of . Then is a polynomial conserved quantity of with .
Using an identical argument to [11, Proposition 2.2], one obtains the following lemma:
Suppose that is a polynomial conserved quantity of . Then the real polynomial has constant coefficients.
From now on we shall assume that is an umbilic-free -surface and assume that is the middle potential .
Suppose that is a degree polynomial conserved quantity of . Then
For the Christoffel dual lifts , one has that . Furthermore, are constants.
For any , has degree at most and the coefficient of is given by .
Consider the polynomial whose coefficients take values in :
Therefore and thus is constant. Furthermore, . Now by Equation (4), in terms of special lifts of the curvature spheres, the middle potential is given by
Thus, implies that
One deduces that , as otherwise one would have that is a section of satisfying , which contradicts that is umbilic-free. Furthermore, from (8) we have that . Therefore, since , takes values in . Thus
for some smooth functions and . By (7), one has that . Therefore, modulo terms in , one has that
Hence, and , as otherwise
would define a section of satisfying , contradicting that is umbilic-free. Returning to the equation and evaluating the terms taking values in , one has that
Since , for any . Therefore,
is a polynomial of degree at most and the coefficient of is . ∎
Suppose that is a polynomial conserved quantity of degree of the middle pencil of . Then for , has degree strictly less than if and only if (or ) and (respectively, ).
Therefore, the coefficient of vanishes if and only if
Since the top term of is given by , we cannot have that and both vanish as this would imply that has degree strictly less than . Therefore, without loss of generality, assume that . Then (10) is equivalent to and , i.e., and . ∎
Suppose that is a polynomial conserved quantity of . Then the degree of is invariant under gauge transformation if and only if is a polynomial of degree .