The Geometry of Riemannian Curvature Radii

We study the geometric structures associated with curvature radii of curves with values on a Riemannian manifold (M, g). We show the existence of sub-Riemannian manifolds naturally associated with the curvature radii and we investigate their properties. In the particular case of surfaces these sub-Riemannian structures are of Engel type. The main character of our construction is a pair of global vector fields f1,f2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1,f_2$$\end{document}, which encodes intrinsic information on the geometry of (M, g).


Introduction 1.From contact elements to curvature radii
The space of contact elements, first introduced by S. Lie (see [Gei08] pages 6-11, or [Mon02] pages 78-80) as a geometric tool for studying differential equations, marks the birth of contact geometry.Let us recall that given a 2-dimensional Riemannian manifold (M, g), the space of (oriented) contact elements of M is the unit tangent bundle U M , i.e. the set of couples (x, v) where x ∈ M , v ∈ T x M and v = 1.Any regular M -valued curve can be lifted to a U M -valued one through the maps There exists a contact distribution ∆ over U M naturally associated to the lifts (1.1).At a point (x, v) ∈ U M such distribution is defined as where we have denoted . This distribution characterizes the images of (1.1), in the sense that a curve (γ, v) ∶ [0, 1] → U M is in the image of one of such lifts if and only if it is tangent to ∆ and γ is a regular curve.In this paper we study a second order generalization of the space of contact elements, in particular, given a Riemannian manifold (M, g), we study the geometric structures associated to the lifts where R g (γ) is the radius of curvature of γ (the definition is recalled below (1.4)).Such second order construction has affinities with Cartan's prolongation of contact structures (see 6.3 of [Mon02]).Recall that, given a Riemannian manifold (M, g) (throughout the whole paper we assume dim(M ) > 1), the geodesic curvature of a regular curve γ ∶ [0, T ] → M is defined as where D t denotes the covariant derivative along γ, and π γ⊥ ∶ T γ(t) M → T γ(t) M denotes the orthogonal projection to { γ(t)} ⊥ .If the geodesic curvature is never vanishing, we can define the radius of curvature of γ, computed with respect to g, as π γ⊥ (D t γ) π γ⊥ (D t γ) . (1.4) With the sole purpose of simplifying the exposition and the expression of certain equations, we study the following modified versions of (1.2) . (1.5) We define the manifold of curvature radii of (M, g), denoted with R(M, g), as the space of triples (x, V, R) such that x ∈ M , R ∈ T x M ∖ {0} and V ∈ {R} ⊥ , V = R .The map (1.5) lifts regular M -valued curves with never vanishing geodesic curvature to R(M, g)-valued ones.The first central result of this work is the following theorem.
Theorem 1.1.Let (M, g) be a smooth n-dimensional Riemannian manifold, and let R(M, g) be the corresponding manifold of curvature radii.There exists a smooth, rank-n distribution, D(M, g), over R(M, g) with the property that a smooth curve (γ, V, R) ∶ [0, 1] → R(M, g) is in the image of one of the lifts (1.5) if and only if it is tangent to D(M, g) and γ is a regular curve.A local basis for D(M, g) is given by n local vector fields {f 1 , . . ., f n }, which can be characterized in terms of the ODEs satisfied by their integral curves: (1.6) j = 3, . . ., n, where {b 3 (x, V, R), . . ., b n (x, V, R)} is a norm-R local orthogonal basis of {R, V } ⊥ .The distribution D(M, g) is bracket generating of step 3, it holds: (1.7) The fields f 1 , f 2 described in equation (1.6) are closely related to the geometry of (M, g).Indeed, as shown in Section 4, many geometric features of the original Riemannian manifold can be recovered considering their Lie brackets.The first bracket [f 1 , f 2 ] gives us back the geodesic flow of (M, g) (see Proposition 4.1); in a way these fields factorize the geodesic flow.Moreover we can read the Riemann curvature tensor in the structure constant of the frame (1.6) (see Proposition 4.3).

Metric structures
The knowledge of the curvature radii of curves of a Riemannian manifold (M, g), is enough to characterize the metric g up to a homothetic transformation, indeed, as shown in Section 3, two Riemannian manifolds having the same curvature radii are homothetic (see Definition 3.2 for the precise statement).
Theorem 1.2.Two Riemannian manifolds (M, g) and (N, η) have the same curvature radii if and only if they are homothetic .
It is then natural to endow the distribution D(M, g) with a metric which is invariant under the action of the homothety group of (M, g).In Section 3 we show the existence of a family of metrics η a,b on D(M, g), parametrized by two real numbers a and b, having this invariance property.The triple (R(M, g), D(M, g), η a,b ) is a sub-Riemannian manifold ([ABB20], [Mon02]), which we denote with R a,b (M, g).For any (γ, R, V ) in the image of the lift (1.5) the metric η a,b satisfies the following equation The central result regarding these metrics is stated in the following theorem.
Theorem 1.3.Let (M, g) be a Riemannian manifold, let a, b ∈ R, b > 0, and let R a,b (M, g) be the corresponding sub-Riemannian manifold of curvature radii.
The following map is a group isomorphism (1.9) ) if and only (N, η) and (M, g) have the same curvature radii.
In Section 5 we show that the sub-Riemannian manifolds R a,b (R 2 , g e ), where g e is the standard Euclidean metric, are all isometric to left-invariant sub-Riemannian structures on the group orientation preserving homothetic transformations of (R 2 , g e ), and we give a characterization of their geodesics.These geometries are related to the left-invariant sub-Riemannian structure on the group of rigid motions of R 2 ([Ard+21], [MS10], [Sac10], [Sac11]).

Structure of the paper
In Section 2 we describe the sub-Riemannian manifold of curvature radii in the 2-dimensional case.In Section 3 we generalize the constructions of Section 2 to an arbitrary Riemannian manifold (M, g) and we prove Theorems 1.1, 1.2, 1.3.In Section 4 we show that taking Lie brackets of the fields f 1 , f 2 , mentioned in the abstract, we can reconstruct the geodesic flow of the original Riemannian manifold and its Riemann curvature tensor.Finally in Section 5 we study the manifold of curvature radii of R 2 endowed with an homothetic invariant metric.

Curvature Radii on surfaces
Let (M, g) be a 2-dimensional Riemannian manifold, which for simplicity we assume to be oriented.Let γ ∶ [0, 1] → M be an arc-leength parametrized curve, then its curvature can be computed as and, provided that k g (γ) is never vanishing, its radius of curvature is (2.2) For every t ∈ [0, 1], the curvature radius of γ at the point γ(t) is a non-zero tangent vector, for this reason we define the manifold of curvature radii of (M, g) as R(M ) = T M ∖ s o , where s o is the zero-section of T M .Every regular curve with non-vanishing curvature can be lifted to a R(M )-valued curve through the radius of curvature map γ ↦ (γ, R g (γ)). (2.3) We would like to find a distribution D(M, g) over R(M ) characterizing the image of such lift, in the sense that a curve (γ, R) 3) if and only if it is tangent to D(M, g) and γ is regular.To build such a distribution at a point (x 0 , R 0 ) ∈ R(M ) we collect all the velocities of all radii of curvature going through this point, and we take the vector space generated by them Since M is oriented we have a complex strcuture where R ⊥g is the unique vector orthogonal to R, positively oriented with it, satisfying R g = R ⊥g g .Proposition 2.1.The collection of vector spaces defined in (2.4) is a smooth Engel distribution over R(M ) (for basic facts on Engel distributions see for instance [Bry+91], chapter 2).Moreover a curve (γ, R) 3) if and only if it is tangent to D(M, g) and γ is a regular curve.A basis for D(M, g) is given by two vector fields f g 1 , f g 2 , which are characterized in terms of the ODEs satisfied by their integral curves as where D t R denotes the covariant derivative of R(t) along the curve x(t).
Proof.It is a special case of Theorem 1.1, which is proved in Section 3.
What is the geometric significance of the fields f g 1 , f g 2 ?It is quite explicit that the integral curves of f g 2 are dialation of R in a fixed fiber, i.e. curves of the kind t ↦ (x 0 , e t R 0 ), whereas if (γ, R) ∶ [0, 1] → R(M ) is an integral curve of f g 1 , as a consequence of Proposition 2.1, R is the radius of curvature of γ, and hence κ g (γ) = 1 R g .On the other hand according to (2.6) hence κ g (γ) is constant.Thus the integral curves of f g 1 are exactly lifts of curves with constant geodesic curvature.Homothetic transformations preserve the curvature radius map.It is then natural to endow D(M, g) with a metric which is invariant under the action of the homothety group of (M, g).For every a, b ∈ R, b ≠ 0, we define a metric η g a,b on D(M, g) by declaring the fields (2.7) an orthonormal frame.A simple computation, which is a particular case of (3.47), shows that if (γ, R) is an admissible curve, then the metric η g a,b satisfies the following equation (2.8) The homogeneity of the right hand side shows the homothetic invariant nature of the metric η g a,b .The triple (R(M ), D(M, g), η g a,b ) defines a sub-Riemannian manifold, which we denote with R a,b (M, g).

The sub-Riemannian manifold of curvature radii
We would like to extend the construction of R a,b (M, g), presented for surfaces in Section 2, to an arbitrary Riemannian manifold.One of the advantages of working with surfaces is that, given a regular curve γ with never vanishing geodesic curvature, the direction of γ is uniquely determined by the radius of curvature.This is no more true when dim(M ) ≥ 3, and we need to keep track of the velocity's direction in some way.For this reason we cannot define the manifold of curvature radii as T M ∖s o anymore.Instead we define it as a subset of T M ⊕ T M .Definition 3.1.Let (M, g) be a Riemannian manifold.The manifold of curvature radii of (M, g), denoted by R(M, g), is defined by (3.1) From now on, when the metric g we are referring to is clear from the context, we drop the g subscript, for instance we denote ⋅ = ⋅ g , ⊥=⊥ g and so on.One can check that, if M is an n-dimensional manifold, then R(M, g) is a smooth embedded sub-manifold of T M ⊕T M of dimension 3n−2.Moreover R(M, g) has the structure of a S n−2 bundle over T M ∖ s o , where the fiber at As expected, if n = 2 these spheres have dimension 0, and R(M, g) reduces to a 0-dimensional fibration over T M ∖ s o .If M is a surface, oriented for simplicity, we have In general R(M, g) has also the structure of a fiber bundle over M , whose fibers are There exist two lifts which allows us to map regular curves γ ∶ [0, 1] → M with never vanishing geodesic curvature, to R(M, g)-valued curves where R g is the map defined in (1.4).We would like to find a distribution characterizing the image of the lift (3.3), i.e. a distribution D(M, g) such that (γ, V, R) ∶ [0, 1] → R(M, g) is the lift of some curve γ if and only if it is tangent to D(M, g), and γ is regular.To construct this distribution at q 0 ∶= (x 0 , V 0 , R 0 ) ∈ R(M, g) we collect the velocities of all such curves going through this point, and we take the vector space generated by them: Before moving to the proof of Theorem 1.1, let us fix the notation to indicate the iterated brackets of the fields f 1 , . . ., f n .Moreover we will often make use of the abbreviation Proof.(Theorem 1.1)The vectorfields in (1.6) are, by definition, local sections of T (T M ⊕ T M ).We need to show that they are actually tangent to be an integral curve of f i with initial point (x 0 , V 0 , R 0 ).We have to show that ⟨V (t), R(t)⟩ = 0 and that V (t) = R(t) for each t such that the flow of f i is defined.Let us start with f 1 , any of its integral curves satisfies Therefore we have Since the condition σ ∈ ⟨{f 1 , . . ., f n }⟩ is independent of the parametrization, without loss of generality we may assume γ = 1, i.e. u 1 = 1 R .Since γ is arc-length parametrized, as a consequence of formulae (1.4) and (1.3), we have On the other hand, since γ = 1 R V , we have where we have denoted To prove that (3.6) holds we just have to show that λ 1 = −u 1 , λ 2 = u 2 .We begin with λ 1 : Concerning the second coefficient λ 2 we have Conversely let σ = (γ, V, R) be a curve satisfying (3.6), with u 1 ≠ 0. We want to show that R = R g (γ), i.e. that R is the radius of curvature of γ.Exploiting (3.6) we compute then, by definition (1.4), the radius of curvature of γ satisfies On the other hand combining (1.3) with (3.8), and considering that R = V , we deduce (3.10) Substituting (3.10) into (3.9)we obtain R g (γ) = R, as requested.
We now show that D(M, g) is an equiregular bracket generating distribution of step 3. To do this we need to be more precise about the definition of the local fields f 3 , . . ., f n in equation (1.6), in particular given (x, V, R) ∈ R(M, g) we have to make a choice of a local basis for {R, V } ⊥ .Let U ⊂ M be an open subset and let E 1 , . . ., E n be a local orthonormal frame for g on U.For every i ≠ j we define the following two-form ω ij ∈ Ω 2 (U): and the following open subset of By construction, the sets to be the norm-R basis of {R, V } ⊥ obtained by applying the Gram-Schmidt algorithm to the vectors {R, V, E 3 (x), . . ., Êi (x), . . ., Êj (x), . . ., E n (x)}, and, on V ij , we set Defined in this way, the local fields e k satisfy two useful properties : for every λ > 0 and for every θ ∈ [0, 2π].Let (x µ ) be local coordinates on M and let (x µ , R µ , V µ ) be the local coordinates induced by (x µ ) on T M ⊕ T M .Let Γ µ αβ be the corresponding Christoffel symbols of the metric g.Given X, Y ∈ R n , we denote with Γ(X, Y ) the row vector then, making use of the notation (3.5), the vector fields f 1 , f 2 read and their commutator reads (3.16) We define the vector field X 12 as (3.17) We define and we notice that this vector field satisfies Finally, to span the whole tangent space, we need the following vector fields from the third layer We start by calculating the field where (x(t), R(t), V (t)) is an integral curve of f 1 .Then, according to (1.6), we have Therefore and hence we have (3.21) Observe that the field f 1k1 satisfies Now we claim that the following 3n − 2 vector fields If we apply π M ⋆ to both sides of (3.24), since according to (1.6), (3.17), (3.21), the fields f 2 , . . ., f n , X 12 , f 13 , . . ., f 1n have x-component equal to zero, we are left with but f 3 , . . ., f n are linearly independent by definition, therefore also We claim that the distribution ⟨{f 2 , . . ., ) and as a consequence therefore, since f 1 , . . ., f n , f 12 , . . ., f 1n are linearly independent, we deduce To prove our claim observe that, for j = 2, . . ., n, [f 2 , f j ] = 0, indeed e tf2 (x, V, R) = (x, e t R, e t V ), hence by construction e j (e tf2 (x, V, R)) = e t e j (x, V, R), or in other words f 2 (e j ) = e j , but then Now consider f j , f i with i, j ≥ 3, to lighten the notation in the following we have denoted e ij (t) = e j (e tfi (x 0 , V 0 , R 0 )), and let us denote a ijk = ⟨ ėij , e k ⟩ R 2 , then we have meaning that and this concludes the proof of the claim.Equation (3.28), together with the fact that the fields {f j } n j=1 ,{f 1k } n k=2 , {f 1k1 } n k=3 are a basis of T R(M, g), gives us the following local characterization of D(M, g)'s flag: (3.30) In particular we have We obtain the following corollary.
Corollary 3.1.The following inclusions hold: Proof.We have already proved the first inclusion of (3.32) since we have shown that ⟨{f 2 , . . ., f n }⟩ is integrable.To prove the second inclusion observe that the kernel of π M ⋆ has dimension (3n − 2) − n = 2n − 2. On the other hand equations (1.6), (3.17), (3.21), imply that the following 2n − 2 linearly independent local smooth sections of D 2 (M, g) f 2 , f 3 , . . ., f n , X 12 , f 13 , . . ., f 1n , (3.33) are contained in ker π M ⋆ , and hence they constitute a basis for it: The distribution ker π M ⋆ is integrable: if X, Y are sections of ker π M ⋆ , then they are π M -related to the zero-section of T M , and therefore also their bracket is so.Thanks to the integrability of ker π M ⋆ , recalling that If (M, g) is a Riemannian surface, which for simplicity we assume to be orientable, then the fields f 1 , f 2 defined in (1.6) can be related with the fields f g 1 , f g 2 defined in (2.6).Indeed, in this case R(M, g) has two connected components: The projection p ∶ R(M, g) → T M ∖ s o restricted to the first of such components is a diffeomorphism satisfying This fact, combined with Theorem 1.1, constitutes a proof of Propositions 2.1.Focusing back on the general case, we would like to define a metric on D(M, g) that preserves the symmetries of the curvature radii lift hence before defining such a metric, we need to have a better understanding of how much information is encoded in the mapping (3.34).
Definition 3.2.Let (M, g) and (N, η) be Riemannian manifolds, we say that they have the same curvature radii if and only if there exists a diffeomorphism ϕ ∶ (M, g) → (N, η) such that This implies that γ ∶ [0, 1] → M is the reparametrization of a geodesic of (M, g) if and only if ϕ ○ γ is the reparametrization of a geodesic of (N, η).In particular (3.36) Definition 3.2 gives a precise meaning to the statement of Theorem 1.2, which we now prove.
Proof.(Theorem 1.2) It is sufficient to prove the theorem for two Riemannian metrics η, g on the same manifold M .Assume firs that (M, g) and (M, η) are homothetic manifolds, then η = λg for some λ > 0. The two metrics have the same Levi-Civita connection ∇, moreover for every X, Y ∈ T M , X ⊥ g Y if and only if X ⊥ η Y , therefore we simply denote ⊥=⊥ g =⊥ η .Let γ ∶ [0, T ] → M be a regular curve, then the curvature computed with respect to η can be easily related to the curvature computed with respect to g: Therefore k η (γ) is never vanishing if and only if also k g (γ) is so, and in that case using equation (1.4) we deduce R η (γ) = R g (γ).
(3.35) holds and for any regular curve γ with never vanishing geodesic curvature.Observe that for any X, Y ∈ T x M , X ⊥ g Y if and only if there exists γ ∶ According to Remark 3.1 we have Hence for any curve γ ∶ [0, 1] → M the following proportionality relationship holds and in particular π γ⊥ D g t γ = π γ⊥ D η t γ. (3.39) In light of (3.39), condition (3.37) reduces to which, by definition of geodesic curvature, in turn is equivalent to which in turn implies that the two metrics are conformally related: there exists a smooth function f ∶ M → R such that η = e 2f g. (3.42) Since the metrics are conformal, their Levi-Civita connection difference tensor can be easily computed On the other hand thanks to equation 3.38 we know that T (X, X) ∝ X, and hence for any Y ⊥ X we have Since X, Y are arbitrary orthogonal vectors, this implies that f is constant.
The curvature radius lift R g of a Riemannian manifold (M, g) is a complete homothety invariant of the metric g.It is then natural to define on D(M, g) a metric which is invariant under homothetic transformations.For every a, b ∈ R, b ≠ 0, we define a metric η a,b on D(M, g) by declaring the fields To prove (3.44) recall that according to Theorem 1.1 there exists u 1 , . . ., or equivalently according to (3.6), . (3.47) The sub-Riemannian manifold (R(M, g), D(M, g), η a,b ) is called the sub-Riemannian manifold of curvature radii of (M, g) and it is denoted with R a,b (M, g).We now prove the main result regarding these metrics, Theorem 1.3.
Proof.(Theorem 1.3)We begin by making sure that the map (1.9) is well defined.Let ϕ ∶ (M, g) → (M, g) be a homothety of Riemannian manifolds.By construction this is a simple consequence of the fact that ϕ ⋆ preserves orthogonality and the ratios between norms of vectors.To prove that Φ ∶= ϕ ⋆ ⊕ ϕ ⋆ is a sub-Riemannian isometry we show that Φ ⋆ f i = f i , i = 1, 2, and that the fields The case of f 2 is analogous.Since ϕ is a homothety it sends local orthogonal basis of T M to local orthogonal basis, and if X, Y ∈ T x M have the same norm, then so do ϕ ⋆ X, ϕ ⋆ Y .Given (x, V, R) ∈ R(M, g), let {e 3 (x, V, R), . . ., e n (x, V, R)} be the local orthogonal basis of {R, V } ⊥ constructed in the proof of Theorem 1.1, then both {e j (ϕ(x), ϕ ⋆ V, ϕ ⋆ R)} n j=3 and {ϕ ⋆ e j (x, V, R)} n j=3 are othogonal basis of {ϕ ⋆ R, ϕ ⋆ V } ⊥ , therefore they are related by an orthogonal transformation: there exists O ∈ O(n − 2) such that ) be an integral curve of f j , for j ≥ 3, we have Hence the fields {Φ ⋆ f 3 , . . ., Φ ⋆ f n } are related to {f 3 , . . ., f n } by an orthogonal transformation.
The map (1.9) is an injective homomorphism by definition.We need to prove that it is also surjective.
be an isometry of sub-Riemannian manifolds, we have to show that there exists ϕ ∶ (M, g) → (N, η) homothety of Riemannian manifolds such that Assume first n > 2 and consider the projection to Notice that the kernel of π M ⋆ has dimension (3n − 2) − n = 2n − 2. On the other hand, as shown in the proof of Theorem 1.1, the equations (1.6), (3.17), (3.21), imply that the following 2n − 2 linearly independent smooth local sections of D 2 (M, g) f 2 , f 3 , . . ., f n , X 12 , f 13 , . . ., f 1n , (3.50) are contained in ker π M ⋆ , and hence they constitute a basis for it: We claim that ker π M ⋆ ⊂ D 2 (M, g) is the unique integrable sub-bundle of D 2 having maximal rank 2n − 2. We have already noticed that ker π M ⋆ is integrable in the proof of Theorem 1.1.Let D ′ ⊂ D 2 (M, g) be an integrable distribution and let X, Y be smooth sections of D ′ .Then there exists some smooth functions α 1 , . . ., α 2n−1 , β 1 , . . ., β 2n−1 such that On the other hand, since which translates to . ., n, are linearly independent mod D 2 (M, g), we have

. , n and setting
we deduce that therefore rank D ′ ≤ n + 1. Observe that it is not possible that rank D ′ = n + 1, because otherwise D ′ would contain D(M, g), contradicting the hypothesis of integrability.We deduce that rank proving our claim.Since Φ is an isometry, it preserves every layer of D(M, g)'s flag, in the sense that Both ker π M ⋆ and D(M, g) are invariant under Φ ⋆ , therefore also their intersection is so, hence and notice that, according to (3.16) and (3.21), ker L = ⟨{f 3 , . . ., f n }⟩.On the other hand, according to 3.55 , or more simply to ker L = Φ ⋆ ker L, thus Φ ⋆ ⟨{f 3 , . . ., f n }⟩ = ⟨{f 3 , . . ., f n }⟩.So far we have deduced that Φ ⋆ ⟨{f 1 , f 3 , . . ., f n }⟩ = ⟨{f 1 , f 3 , . . ., f n }⟩, taking the orthonormal complement of this last equation with respect to the sub-Riemannian metric, we deduce that Φ therefore the following map is well defined diffeomorphism but we know that Φ x = ϕ, hence the first equation of (3.57) reads recall that, according to (3.16), its integral curves satisfy on the other hand since Exploiting (3.59) together with the third equation in (3.57) we deduce Substituting (3.60) in the second equation of (3.57) we find that we have to discard the minus sign and we are left with ) be a curve tangent to D(M, g) with γ regular, then according to Theorem 1.1, R = R g (γ).On the other hand since Φ preserves D(M, g), also (ϕ ○ γ, ϕ ⋆ R, ϕ ⋆ V ) is tangent to D(M, g) and hence, according to Theorem 1.1 we have On the other hand, since ϕ ∶ (M, ϕ ⋆ g) → (M, g) is an isometry, it holds Assume now that n = dim M = 2. Consider the following linear map Observe that the kernel of (3.62), is invariant under pushforwards of sub-Riemannian isometries, in the sense that and since Φ is an isometry we deduce that We claim that Φ ⋆ f 2 = f 2 .Assume by contradiction that Φ ⋆ f 2 = −f 2 and consider the following couple of vector fields On the other hand since The remaining part of the proof is identical to the one of the case n > 2. In particular we can conclude by repeating verbatim what is written between equation (3.56) and the discussion of the 2-dimensional case.
4 The fields f 1 , f 2 Given a Riemannian manifold (M, g), the vector fields f 1 , f 2 defined in (1.6) are global sections of T (T M ⊕ T M ), which restrict to vector fields on R(M, g).As a consequence of Theorem 1.1 we know that they are metric invariants of (M, g).Actually they are even invariant under homothetic transformation.In the current section we show that some classical metric invariants can be recovered from the iterated Lie brackets of f 1 , f 2 .Indeed, as the next proposition shows, the field [f 1 , f 2 ] already gives us a complete description of the geodesics of (M, g); in this sense the fields f 1 , f 2 give us a factorization of the geodesic flow.
Proposition 4.1.Let (M, g) be a Riemannian manifold, let x ∈ M and let exp (M,g) x be the corresponding exponential map at x. Let f 21 = [f 2 , f 1 ] and for every t ∈ R such that the flow of the fields is defined.
Proof.According to equation (3.16), the vector field f 21 reads where the symbols Γ are the ones defined in equation (3.14).Consequently, any integral curve of f 21 satisfies Therefore the curve x(t) = π M ○ e t[f2,f1] (x 0 , V 0 , R 0 ) is the unique geodesic with initial point x 0 and initial velocity V 0 .
Recall the vector field X 12 defined in equation (3.17) and observe that therefore Hence any integral curve of f 121 satisfies concluding the proof.
If we consider another layer of the Lie algebra generated by f 1 , f 2 , the components of Riemann curvature tensor appear.Proposition 4.2.Let (M, g) be a Riemannian manifold, the integral curves of the vector field Proof.Given a vector field X we denote with X x its x-component, with X R its R-component and with We compute (4.4) component by component: (4.5) Concerning the R-component we compute and (4.7) Equations (4.5), (4.6) and (4.7) together imply that any integral curve of f 1121 satisfies the ODEs system (4.4).
As stated in Theorem 1.1, the fields {f 1 , . . ., f n , f 12 , . . ., f 1n , f 121 , . . ., f 1n1 } (4.8) constitute a local basis for T R(M, g), therefore there exists some real valued smooth functions {c 1 , . . ., c 3n−2 } on R(M, g) such that (4.9) Proposition 4.3.Let (M, g) be a Riemannian manifold, then the function c 1 ∶ R(M, g) → R defined in equation (4.9) is a homothetic invariant of (M, g), which can be expressed in terms of sectional curvatures as In particular if (M, g) is a Riemannian surface, then where K is the Gaussian curvature of (M, g).
Proof.From equations (4.4) and (4.2) it follows that where {e 3 , . . ., e n } is the basis of {R, V } ⊥ described in the proof of Theorem 1.1 and we have denoted The unique expression of the vector field X in terms of the basis (4.8) is a linear combination which does not involve f 1 , hence we deduce The fields f 1 , f 2 allow us to characterize the homotheties of Riemannian surfaces with one synthetic equation.Let M be a smooth manifold and let X ∈ X(M ) be a vector field, which we assume to be complete for simplicity.The family of maps {e tX ⋆ } t∈R constitutes a one-parameter group of diffeomorphisms of T M ; we denote its infinitesimal generator with #» X.
Proposition 4.4.Let (M, g) be a Riemannian surface and let f 1 be the local vector vield over T M ∖ s o defined in (2.6).A vector field X ∈ X(M ) is the infinitesimal generator of a one-parameter group of homotheties if and only if

.15)
Proof.A vector field X ∈ X(M ) satisfies (4.15) if and only if for every t ∈ R. On the other hand, any vector field X ∈ X(M ) satisfies Therefore X satisfies (4.15) if and only if e t #» X is an isometry of R a,b (M, g) for each t ∈ R, hence, by Theorem 1.3, X satisfies (4.15) if and only if e tX is a one-parameter group of homotheties of (M, g).
The results obtained so far can be used to produce a flatness theorem for Riemannian surfaces having a 4-dimensional Lie algebra of homothetic vector fields.
Theorem 4.1.Let (M, g) be a 2-dimensional Riemannian manifold and let L ⊂ X(M ) be the corresponding Lie algebra of homothetic vector fields.Then dim(L) ≤ 4, and (M, g) is flat if equality is achieved.
Proof.Let L ⊂ X(R(M )) be the Lie algebra of isometric vector fields for the manifold of curvature radii R(M, g).By Theorem 1.3 this Lie algebra is isomorphic to the one of homothetic vector fields of (M, g).Let X ∈ L be a vector field vanishing at some point (x, R) ∈ R(M ).We claim that X is identically zero.Indeed, since X is isometric, by Proposition 4.4, it satisfies Let X 1 , . . ., X 5 ∈ L and (x, R) ∈ R(M ).The manifold of curvature radii is 4-dimensional, hence there exists a linear combination of X 1 , . . ., X 5 vanishing at (x, R), and thus vanishing everywhere.We deduce that dim L ≤ 4. Assume now that dim L = 4 and let X 1 , . . ., X 4 be a basis for L. By the above argument X 1 , . . ., X 4 are linearly independent at every point of R(M ).Thus they can be used to produce local coordinates in the neighbourhood of every point, by means of the map (s 1 , . . ., s 4 ) ↦ e s1X1 ○ ⋅ ⋅ ⋅ ○ e s4X4 (x, R).
This implies that the structure coefficients of the frame f 1 , f 2 , f 12 , f 121 are constants (Theorem 1.1 implies that the latter is a basis for T R(M )).In particular there exist real constants c 1 , c 2 , c 12 , c 121 such that is the Gaussian curvature of (M, g).Such a function is constant on a fixed fiber of the manifold of curvature radii R(M ) = T M ∖ s 0 if and only if it is identically zero.It follows that K = 0.

Similarity transformations of the plane
In this section we show that the sub-Riemannian manifolds R a,b (R 2 , g e ), where g e is the standard Euclidean metric, are isomorphic to left invariant sub-Riemannian structures on the group of orientation preserving homotheties of R 2 , which we denote with G. Moreover we give a a characterization of the sub-Riemannian geodesics of R 0,1 (R 2 , g e ) in terms of the euclidean curvature of their projections to a plane.Such characterization in terms of curvature is similar to the one of Euler elasticae ( [Lev]), which are projection of normal extremal trajectories of the nilpotent Engel group ([AS11]).All the results of this section follow from straightforward computations, therefore the proofs are omitted.Let (y 1 , y 2 , R 1 , R 2 ) be global coordinates for T R 2 ∖ {0} = R 2 × R 2 ∖ {0}.It is convenient to define a new set of coordinates (θ, r, x 1 , x 2 ) as (y 1 , y 2 , R 1 , R 2 ) =∶ (x 1 + rcosθ, x 2 + rsinθ, −rcosθ, −rsinθ). (5.1) In coordinates (y, R) a point in R a,b (R 2 , g e ) is interpreted as the curvature radius of some curve going through y.In the new coordinates (θ, r, x 1 , x 2 ) we interpret a point in R a,b (R 2 , g e ) as the osculating circle of such curve, having radius (r cos θ, r sin θ) and center (x 1 , x 2 ).Observe that the data of a homothetic transformation, which in the case of (R 2 , g e ) consists in a composition of rotations, dialations and translation, can be encoded into an osculating circle, i.e. a point of R a,b (R 2 , g e ): given a circle (θ, r, x 1 , x 2 ) we can dilate by r, rotate by θ and translate by (x 1 , x 2 ).We have a diffeomorphism which allows us to push forward the sub-Riemannian structure of R a,b (R 2 , g e ), to G. The resulting sub-Riemannian structure is left invariant.
Proposition 5.1.The frame (f 1 , f 2 ) can be written in coordinates (θ, r, x 1 , x 2 ) as (5.3) Both of these vector fields are pushforwarded to left invariant vector fields by the map (5.2): (5.4) Remark 5.1.The geometry of G is reminiscent of the left invariant sub-Riemannian structure on the group of rigid motions of the plane, related to 'bicycling mathematics', which has been studied in [Ard+21], [MS10], [Sac10], [Sac11].The group of rigid motions of R 2 can be described as In coordinates (θ, x 1 , x 2 ) we can define a left-invaraint sub-Riemann structure on SE 2 by declaring the fields an orthonormal generating family.There exists a submersion P ∶ G → SE 2 (θ, r, x 1 , x 2 ) ↦ (θ, x 1 , x 2 ), (5.6) (5.7) Let h 1 , h 2 ∶ T ⋆ R(R 2 ) → R be the Hamiltonian functions associated with f 1 , f 2 and let p θ , p r , p x1 , p x2 be the canonical momenta associated with the coordinates θ, r, x 1 , x 2 , then (5.9) The next proposition characterizes normal extremal trajectories in terms of the euclidean curvature of their projection to the (ρ, θ)-plane.There are no strictly abnormal extremals.
frame.For any (γ, R, V ) in the image of the lift (3.3) the metric η a,b satisfies the following equation