Sub-Riemannian curvature and a Gauss–Bonnet theorem in the Heisenberg group
We use a Riemannnian approximation scheme to define a notion of sub-Riemannian Gaussian curvature for a Euclidean -smooth surface in the Heisenberg group away from characteristic points, and a notion of sub-Riemannian signed geodesic curvature for Euclidean -smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss–Bonnet theorem. An application to Steiner’s formula for the Carnot-Carathéodory distance in is provided.
Key words and phrases:Heisenberg group, sub-Riemannian geometry, Riemannian approximation, Gauss–Bonnet theorem, Steiner formula
2010 Mathematics Subject Classification:Primary 53C17; Secondary 53A35, 52A39
- 1 Introduction
- 2 Notation and background
- 3 Riemannian approximation of curvature of curves
- 4 Riemannian approximation of geodesic curvature of curves on surfaces
- 5 Riemannian approximation of curvatures of surfaces
- 6 A Gauss–Bonnet theorem
- 7 Application: a simplified Steiner’s formula
- 8 Questions and remarks
- A Examples
A full understanding of the notion of curvature has been at the core of studies in differential geometry since the foundational works of Gauss and Riemann. The aim of this paper is to propose a suitable candidate for the notion of sub-Riemannian Gaussian curvature for Euclidean -smooth surfaces in the first Heisenberg group , adopting the so called Riemannian approximation scheme, which has proved to be a very powerful tool to address sub-Riemannian issues.
Referencing the seminal work of Gauss, we recall that to a compact and oriented Euclidean -smooth regular surface we can attach the notions of mean curvature and Gaussian curvature as symmetric polynomials of the second fundamental form. To be more precise, for every we have a well-defined outward unit normal vector field, , usually called the Gauss normal map. For every , the differential of the Gauss normal map defines a positive definite and symmetric quadratic form on whose two real eigenvalues are usually called principal curvatures of at . The arithmetic mean of these principal curvatures is the mean curvature and their product is the Gaussian curvature. The importance of the latter became particularly clear after Gauss’ famous Theorema Egregium, which asserts that Gaussian curvature is intrinsic and is also an isometric invariant of the surface .
The notions of curvature, as briefly recalled above, can be extended to far more general situations, for instance to submanifolds of higher codimension in , and also to the broader geometrical context provided by Riemannian geometry, as was done by Riemann. In particular, we will consider 2-dimensional Riemannian manifolds isometrically embedded into 3-dimensional Riemannian manifolds. We refer to Section 5 for details.
Our interest in the study of curvatures of surfaces in is motivated by the still ongoing studies in the context of sub-Riemannian manifolds or more specific structures like Carnot groups, whose easiest example is provided by the first Heisenberg group . Restricting our attention to , there is a currently accepted notion of horizontal mean curvature at non-characteristic points of Euclidean regular surfaces. This notion has been considered by Pauls () via the method of Riemannian approximants, but has also been proved to be equivalent to other notions of mean curvature appearing in the literature (e.g.  or ).
The method of Riemannian approximants relies on a famous result due to Gromov, which states that the metric space can be obtained as the pointed Gromov-Hausdorff limit of a family of metric spaces , where is a suitable family of Riemannian metrics. The Riemannian approximation scheme has also proved to be a very efficient tool in more analytical settings, for instance, in the study of estimates for fundamental solutions of the sub-Laplacian (e.g. [18, 8]) as well as regularity theory for sub-Riemannian curvature flows (e.g. ). The preceding represents only a small sample of the many applications of the Riemannian approximation method in sub-Riemannian geometric analysis, and we refer the reader to the previously cited papers for more information and references to other work in the literature. The monograph  provides a detailed description of the Riemannian approximation scheme in the setting of the Heisenberg group.
Let us denote by and the left-invariant vector fields which span the Lie algebra of . In particular, . In order to exploit the contact nature of it is customary to define an inner product which makes an orthonormal basis. A possible way to define a Riemannian scalar product is to set for every , and then to extend to a scalar product which makes an orthonormal basis. The family of metric spaces converges to in the pointed Gromov-Hausdorff sense.
Within this family of Riemannian manifolds, we can now perform computations adopting the unique Levi-Civita connection associated to the family of Riemannian metrics . Obviously, all the results are expected depend on the positive constant . The plan is to extract horizontal notions out of the computed objects and to study their asymptotics in as . This is the technique adopted in  to define a notion of horizontal mean curvature.
It is natural to ask whether such a method can be employed to study the curvature of curves, and especially to articulate an appropriate notion of sub-Riemannian Gaussian curvature. One attempt in this direction has been carried out in , where the authors proposed a notion of horizontal second fundamental form in relation with -convexity. A different notion of sub-Riemannian Gaussian curvature for graphs has been suggested in .
Our approach follows closely the classical theory of Riemannian geometry and leads us to the following notion of sub-Riemannian curvature for a Euclidean -smooth and regular curve :
Here is the standard contact form on . We stress that, when dealing with purely horizontal curves, the above notion of curvature is already known and appears frequently in the literature.
An analogous procedure allow us to define also a notion of sub-Riemannian signed geodesic curvature for Euclidean -smooth and regular curves living on a surface , with . This notion takes the form
In the same spirit we introduce a notion of sub-Riemannian Gaussian curvature away from characteristic points. We will work with Euclidean -smooth surfaces , whose characteristic set is defined as the set of points where . The explicit expression of reads as follows:
The quantity in (1.3) cannot easily be viewed as a symmetric polynomial of any kind of horizontal Hessian. Moreover, the expression of written above resembles one of the integrands, the one which would be expected to replace the classical Gaussian curvature, appearing in the Heisenberg Steiner’s formula proved in . The discrepancy between these two quantities will be the object of further investigation.
The definition of an appropriate notion of sub-Riemannian Gaussian curvature leads to the question of proving a suitable Heisenberg version of the celebrated Gauss–Bonnet Theorem, which is the first main result of this paper. For a surface with , our main theorem is as follows.
Let be a regular surface with finitely many boundary components , , given by Euclidean -smooth regular and closed curves Let be the sub-Riemannian Gaussian curvature of , and the sub-Riemannian signed geodesic curvature of relative to . Suppose that the characteristic set satisfies , and that is locally summable with respect to the Euclidean 2-dimensional Hausdorff measure near the characteristic set . Then
The sharpness of the assumption made on the 1-dimensional Euclidean Hausdorff measure of the characteristic set is discussed in Section 6, while comments on the local summability asked for are postponed to Section 8. The measure on the th boundary curve in the statement of Theorem 1.1 is the limit of scaled length measures in the Riemannian approximants. We remark that this measure vanishes along purely horizontal boundary curves.
Gauss–Bonnet type theorems have previously been obtained by Diniz and Veloso  for non-characteristic surfaces in , and by Agrachev, Boscain and Sigalotti  for almost-Riemannian structures. We would also like to mention the results obtained by Bao and Chern  in Finsler spaces.
The notion of horizontal mean curvature has featured in a long and ongoing research program concerning the study of constant mean curvature surfaces in , especially in relation to Pansu’s isoperimetric problem (e.g. , , ,  or ). A simplified version of the aforementioned Gauss–Bonnet Theorem 1.1, i.e., when we consider a compact, oriented, Euclidean -surface with no boundary, or with boundary consisting of fully horizontal curves, ensures that the only compact surfaces with constant sub-Riemannian Gaussian curvature have .
The structure of the paper is as follows. In Section 2 we provide a short introduction to the first Heisenberg group and the notation which we will use throughout the paper, with a special focus to the Riemannian approximation scheme. In Section 3 and 4 we adopt the Riemannian approximation scheme to derive the expression (1.1) for the sub-Riemannian curvature of Euclidean -smooth curves in , and the expression (1.2) for the sub-Riemannian geodesic curvature of curves on surfaces. In Section 5, we will derive the expression (1.3) for the sub-Riemannian Gaussian curvature. In Section 6 we prove Theorem 1.1 and its corollaries. Section 7 contains the proof of Steiner’s formula for non-characteristic surfaces. In Section 8 we present a Fenchel-type theorem for horizontal closed curves (see Theorem 8.5) and we pose some questions. One of the more interesting and challenging questions concerns the summability of the sub-Riemannian Gaussian curvature with respect to the Heisenberg perimeter measure near isolated characteristic points. This summability issue is closely related to the open problem posed in  concerning the summability of the horizontal mean curvature with respect to the Riemannian surface measure near the characteristic set. To end the paper, we add an appendix where we collect several examples of surfaces in which we compute explicitly the sub-Riemannian Gaussian curvature .
Research for this paper was conducted during visits of the second and third authors to the University of Bern in 2015 and 2016. The hospitality of the Institute of Mathematics of the University of Bern is gratefully acknowledged. The authors would also like to thank Luca Capogna for many valuable conversations on these topics and for helpful remarks concerning the proof of Theorem 1.1.
2. Notation and background
Let be the first Heisenberg group where the non-commutative group law is given by
The corresponding Lie algebra of left-invariant vector fields admits a -step stratification, , where and for , and . On we consider also the standard contact form of
The left-invariant vector fields and play a major role in the theory of the Heisenberg group because they span a two-dimensional plane distribution , known as the horizontal distribution, which is also the kernel of the contact form :
This smooth distribution of planes is a subbundle of the tangent bundle of , and it is a non integrable distribution because . We can define an inner product on , so that for every , forms a orthonormal basis of . We will then denote by the horizontal norm induced by the scalar product . In both cases, we will omit the dependence on the base point when it is clear.
An absolutely continuous curve is said to be horizontal if for a.e. .
Let a horizontal curve. The horizontal length of is defined as
It is standard to equip the Heisenberg group with a path-metric known as Carnot-Carathéodory, or , distance:
Let , with . The distance between is defined as
Dilations of the Heisenberg group are defined as follows:
It is easy to verify that dilations are compatible with the group operation: , , , and that the distance is homogeneous of order one with respect to dilations: . The scaling behavior of the left-invariant vector fields with respect to dilations is as follows:
We are now ready to implement the Riemannian approximation scheme. First, let us define for . We define a family of Riemannian metrics on such that becomes an orthonormal basis. The choice of this specific family of Riemannian metrics on is indicated by the following theorem.
Theorem 2.4 (Gromov).
The family of metric spaces converges to in the pointed Gromov-Hausdorff sense as .
This deep result continue to hold even for more general Carnot groups, but there is one additional feature which is valid for :
Any length minimizing horizontal curve joining to the origin is the uniform limit as of geodesic arcs joining to in the Riemannian manifold .
For both results, we refer to [10, Chapter 2].
Continuing with notation, the scalar product that makes an orthonormal basis will be denoted by . Explicitly, this means that, given and ,
Obviously, if we write and in the basis, i.e., where (and similarly for ), we have
The following relations allow us to switch from the standard basis to and vice versa:
In exponential coordinates, the metric is represented by the symmetric matrix , for In particular,
Following the classical notation of Riemannian geometry, we
will denote by the elements of the matrix , and by
the elements of its inverse .
A standard computational tool in Riemannian geometry is the notion of affine connection.
Let be the set of -smooth vector fields on a Riemannian manifold . Let be the ring of real-valued -smooth functions on . An affine connection on is a mapping
usually denoted by , such that:
for every and for every .
It is well known that every Riemannian manifold is equipped with a privileged affine connection: the Levi-Civita connection . This is the unique affine connection which is compatible with the given Riemannian metric and symmetric, i.e.,
for every . A direct proof of this fact yields the famous Koszul identity:
It is possible to write the Levi-Civita connection in a local frame by making use of the Christoffel symbols . In our case, due to the specific nature of the Riemannian manifold , we can use a global chart given by the identity map of . The Christoffel symbols are uniquely determined by
The Christoffel symbols of the Levi-Civita connection of are given by
It is a direct computation using
for . ∎
We now compute the Levi-Civita connection associated to the Riemannian metric .
The action of the Levi-Civita connection of on the vectors , and is given by
It follows from a direct application of the Koszul identity (2.2), which here simplifies to
To make the paper self-contained, we recall here the definitions of Riemann curvature tensor and of sectional curvature.
The Riemann curvature tensor of a Riemannian manifold is a mapping defined as follows
Note that the Riemann curvature tensor satisfies the functional property
for every and every .
Let be a Riemannian manifold and let be a two-dimensional subspace of the tangent space . Let be two linearly independent vectors in . The sectional curvature of is defined as
where denotes the usual wedge product.
One of the main reasons to introduce the notion of affine connection, is to be able to differentiate smooth vector fields along curves: this operation is known as covariant differentiation (see , Chapter 2). Formally, let be a smooth vector field (written in the standard basis of ) along a curve . The covariant derivative of along the curve is given by
where the ’s are the coordinates of in a local chart and are the Christoffel symbols introduced before.
In particular, if , we have
3. Riemannian approximation of curvature of curves
Let us define the objects we are going to study in this section.
Let be a Euclidean -smooth curve. We say that is regular if
Moreover, we say that is a horizontal point of if
Let be a Euclidean -smooth regular curve in the Riemannian manifold . The curvature of at is defined as
We stress that the above definition is well posed, indeed by Cauchy-Schwarz,
We recall that in Riemannian geometry the standard definition of curvature for a curve parametrized by arc length is . The one we gave before is just more practical to perform computations for curves with an arbitrary parametrization.
Let us briefly recall the definition of -geodesics, cf. [10, Chapter 2]. For a Euclidean -smooth regular curve , define its penalized energy functional to be
Using a standard variational argument, we can derive the system of Euler-Lagrange equations for the functional : we will call -geodesics the critical points, which are actually curves, of the functional . In other words, we will say that is a -geodesic, if for every it holds that
We are now ready to present the first result concerning the curvature .
Let be a Euclidean -smooth regular curve in the Riemannian manifold . Then
In particular, if is a horizontal point of ,