Any isometry between the spheres of absolutely smooth 2 -dimensional Banach spaces is linear
aa r X i v : . [ m a t h . F A ] N ov ANY ISOMETRY BETWEEN THE SPHERES OF SUFFICIENTLY SMOOTH -DIMENSIONAL BANACH SPACES IS LINEAR TARAS BANAKH
Abstract.
We introduce a class of sufficiently smooth Banach spaces X (which includes all C -smooth 2-dimensional Banach spaces) and prove that any isometry f : S X → S Y between theunit spheres of sufficiently smooth Banach spaces X, Y extends to a linear isometry ¯ f : X → Y of the Banach spaces X, Y . This answers the famous Tingley’s problem in the class of sufficientlysmooth 2-dimensional Banach spaces. Introduction
According to a classical result of Mazur and Ulam [11] (extended by Mankiewicz [10]), everybijective isometry f : B X → B X between the unit balls of Banach spaces X, Y extends to alinear bijective isometry of the spaces
X, Y . Tingley [16] asked in 1987 if the unit balls in thisMazur–Ulam–Mankiewicz result can be replaced by the unit spheres:
Problem 1.1 (Tingley) . Let f : S X → S Y be a bijective isometry between the unit spheres ofBanach spaces X, Y . Can f be extended to a linear isometry of the Banach spaces X, Y ? Here for a Banach space ( X, k · k ) by B X = { x ∈ X : k x k ≤ } and S X = { x ∈ X : k x k = 1 } we denote the unit ball and the unit sphere of X , respectively.Motivated by the Tingley Problem, the following property of Banach spaces was introduced byCheng and Dong [1], and then widely used in the literature devoted to Tingley’s problem. Definition 1.2.
A Banach space X has the Mazur-Ulam property if every bijective isometry f : S X → S Y from the unit sphere of X to the unit sphere of any Banach space Y extends to alinear isometry between the Banach spaces X, Y .So, the Tingley Problem 1.1 has an affirmative solution if and only if all Banach spaces have theMazur-Ulam property. Many classical Banach spaces do have the Mazur-Ulam property, see thesurveys [2], [12], [18] and references therein. According to [6], every polyhedral finite-dimensionalBanach space has the Mazur-Ulam property. Surprisingly, but it is still not known if every 2-dimensional Banach space has the Mazur-Ulam property. The latter problem was discussed in thepapers [3], [13], [14], [17]. A substantial progress in resolving the 2-dimensional case of the Tingleyproblem was made by Javier Cabello S´anchez who proved in [13] that an isometry f : S X → S Y between the spheres of 2-dimensional Banach spaces extends to a linear isometry between X and Y if and only if f is linear on some nonempty relatively open subset U ⊂ S X . The linearity of f on U means that f ( ax + by ) = a · f ( x ) + b · f ( y ) for any x, y ∈ U and any real numbers a, b with ax + by ∈ U . Generalizing the result of Wang and X. Huan [17], S´anchez also proved in [13, 3.8]that a two-dimensional Banach space X has the Mazur-Ulam property if its unit ball B X is notstrictly convex. Therefore, the 2-dimensional case of the Tingley Problem remain open only forstrictly convex Banach spaces. It is also open for smooth Banach spaces. Let us recall [ ? ] that aBanach space X is smooth if for each x ∈ S X there exists a unique linear continuous functional x ∗ : X → R such that x ∗ ( x ) = 1 = k x ∗ k . Geometrically this means that the unit ball B X has aunique supporting hyperplane at x .In this paper we shall present a solution of the Tingley Problem 1.1 in a subclass of smooth2-dimensional Banach spaces consisting of sufficiently smooth Banach spaces. Sufficiently smooth Mathematics Subject Classification.
Key words and phrases.
Banach space, isometry, sphere, natural parametrization.
Banach spaces are introduced in Definition 1.4 below. But first we need to recall some notionsfrom Real Analysis.Let ( X, k · k ) be a Banach space and k be a positive integer. A function r : U → X defined onan open subset U ⊂ R is • of bounded variation if there exists a real number E such that for any points x < y A 2-dimensional Banach space ( X, k · k ) is defined to be C k -smooth (resp. AC -smooth, AC L -smooth, A ˘ C -smooth ) if there exists a C k -smooth (resp. AC -smooth, AC L -smooth, A ˘ C -smooth) surjective map r : R → S X ⊂ X such that k r ′ ( s ) k = 1 for all s ∈ R .For any 2-dimensional Banach space X these smoothness properties relate as follows. C -smooth + AC L -smooth + AC -smooth + C -smooth k s + smooth A ˘ C -smooth K S + strictly convex Definition 1.4. A 2-dimensional Banach space is defined to be sufficiently smooth if it is AC L -smooth or A ˘ C -smooth.Observe that each C -smooth 2-dimensional Banach space is sufficiently smooth and each suf-ficiently smooth 2-dimensional Banach space is AC -smooth and smooth.The main result of this paper is the following theorem that yields an affirmative answer to theTingley problem for sufficiently smooth 2-dimensional Banach spaces. Theorem 1.5. Each isometry f : S X → S Y between the unit spheres of two sufficiently smooth -dimensional Banach spaces extends to a linear isometry ¯ f : X → Y between the Banach spaces X, Y . We do not know if the sufficient smoothness of the Banach spaces in Theorem 1.5 can beweakened to their AC -smoothness.The proof of Theorem 1.5 essentially uses the sufficient smoothness of both Banach spaces.This motivates the following natural problem. Problem 1.6. Has each sufficiently smooth -dimensional Banach space the Mazur-Ulam prop-erty? Theorem 1.5 reduces Problem 1.6 to the following problem. Problem 1.7. Let f : S X → S Y be an isometry between the unit spheres of -dimensional Banachspaces X and Y . Does the sufficient smoothness of the Banach space X implies the sufficientsmoothness of the Banach space Y ? SOMETRIES BETWEEN SMOOTH 2-DIMENSIONAL BANACH SPACES 3 Theorem 1.5 will be proved in Section 9 after long preparatory work made in Sections 4–8. InSection 3 we introduce the notion of a natural parametrization of the unit sphere of a smooth2-dimensional Banach space and prove that such a parametrization is unique up to an isometricshift in the domain. In Section 4, given a 2-dimensional Banach space with a fixed basis (suchBanach spaces are called 2-based), we introduce the polar parametrization p of the unit sphere S X of X and establish some smoothness properties of this parametrization. In Section 5 we modify thepolar parametrization p of S X to the natural parametrization r of S X , which is a parametrizationwhose tangent vector has length 1 almost everywhere. In Section 6 we introduce the radial andtangential curvatures of the sphere S X . Those are functions ρ and τ such that ρ · r + τ · r ′ + r ′′ = 0.In Theorem 6.2 we prove that the radial and tangential curvatures determine an AC -smooth 2-dimensional Banach space uniquely up to an insometry, and will characterize 2-dimensional Hilbertspaces as the unique smooth Banach spaces with constant radial and tangential curvatures (equalto 1 and 0, respectively). In Section 7, for a smooth 2-based Banach space X we introducethe radial and tangential supercurvatures P and T , which are continuous functions such that P · r + T · r = − r ′ ◦ ϕ , where ϕ : R → R is a continuous non-decreasing function such that r ′ = r ◦ ϕ .We observe that the function τρ extends to the continuous function ψ = TP , called the quotientcurvature of X . In Section 8 we prove that for an AC -smooth 2-based Banach space the radialcurvature ρ is uniquely determined by the metric of the sphere S X . Next, we show that for asufficiently smooth 2-based Banach space the derivative ψ ′ of the quotient curvature ψ is uniquelydetermined by the metric of the sphere S X . Then we apply one result of Floquet Theory onperiodic solutions of second order differential equations to recover the quotient curvature ψ fromits derivative ψ ′ . Knowing the radial and quotient curvatures we finally calculate the tangentialcurvatures τ = ρ · ψ . Thus we conclude that the radial and tangential curvatures of a sufficientlysmooth Banach space X can be uniquely recovered from the metric of the unit sphere. Togetherwith the Uniqueness Theorem 6.2 this yields the proof of Theorem 1.5, presented in the finalSection 9. 2. Some tools from Real Analysis Throughout the paper we shall heavily some standard tools of Real Analysis. In this sectionwe recall or establish some facts from the Real Analysis that will be used in the sequel.By Lebesgue Theorems 7.1.13 and 7.1.15 [7], each locally absolutely continuous function f : R → X is differentiable almost everywhere and f ( b ) − f ( a ) = R ba f ′ ( x ) dx for any a < b . Moreover, R ba | f ′ ( x ) | dx < ∞ . So, a locally absolutely continuous function can be recovered from its derivative(which is a locally integrable function). Let us recall that a function f : R → R is locally integrable if it is measurable and R ba | f ( x ) | dx is finite for any a < b .A point x ∈ R is called the Lebesgue point of a locally integrable function f : R → R iflim ε → ε Z ε | f ( t ) − f ( x ) | dt = 0 . By the classical Lebesgue Theorem [7, 7.1.20 and 7.1.21], almost every point of the real line is aLebesgue point of any locally integrable function f : R → R , and each Lebesgue point of f is adifferentiability point of the function F ( x ) = R x f ( t ) dt .We define a function f : R + → R defined on the half-line R + = (0 , ∞ ) to be of order o [1] iflim ε → ε R ε | f ( u ) | du = 0. It is easy to see that for any functions f, g ∈ R + → R of order o [1] andany real constant c the functions f + g and c · f are of order o [1].For each Lebesgue point x of a locally integrable function f : R → R , the functions f ( x + ε ) − f ( x )and f ( x − ε ) − f ( x ) are of order o [1]. Lemma 2.1. If a locally integrable function f : R + → R is of order o [1] , then the function R u f ( v ) v dv is of order o ( u ) .Proof. Given any ε > δ > (cid:12)(cid:12) R u f ( v ) v dv (cid:12)(cid:12) < εu for any u ∈ (0 , δ ].Since f is of order o [1], there exists δ > R u | f ( v ) | dv ≤ εu for every u ∈ (0 , δ ]. Then TARAS BANAKH for any u ∈ (0 , δ ] we have (cid:12)(cid:12)(cid:12) Z u f ( v ) v dv (cid:12)(cid:12)(cid:12) ≤ Z u | f ( v ) |· v dv ≤ u Z u | f ( v ) | dv < u ( εu ) = εu . (cid:3) We say that a function f : R + → R is of order o [ u ] if the function f ( u ) u is of order o [1].Lemma 2.1 implies the following lemma that will be used in the proof of Lemma 8.2. Lemma 2.2. If a locally integrable function f : R + → R is of order o [ u ] , then the function R u f ( v ) dv is of order o ( u ) . For a positive real number L , a function f : R → R is called L -periodic if f ( x + L ) = f ( x ) forall x ∈ R . We shall need the following (known) property of L -periodic functions. Lemma 2.3. If a locally integrable function f : R → R is L -periodic, then for any a ∈ R , Z a + La f ( t ) dt = Z L f ( t ) dt. Proof. Find a unique integer number n such that nL ≤ a < ( n + 1) L and put b = a − nL . Then Z a + La f ( t ) dt = Z n + L + nLb + nL f ( t ) dt = Z b + Lb f ( t + nL ) dt = Z b + Lb f ( t ) dt = Z Lb f ( t ) dt + Z b + LL f ( t ) dt = Z Lb f ( t ) dt + Z b f ( t + L ) dt = Z Lb f ( t ) dt + Z b f ( t ) dt = Z L f ( t ) dt. (cid:3) A natural parametrization of the sphere of a -dimensional Banach space Let ( X, k · k ) be a smooth 2-dimensional Banach space. By a natural parametrization of thesphere S X = { x ∈ X : k x k = 1 } we understand any C -smooth map r : R → S X such that k r ′ ( s ) k = 1 for every s ∈ R . In this section we shall show that a natural parametrization isunique up to an isometry of the real line. To prove this fact we investigate the relation of naturalparametrizations to the intrinsic metrics on connected subsets of S X .First we recall the necessary information on intrinsic distances. Let ( M, d ) be a metric spaceand ε be a positive real number. A sequence of points x , . . . , x n ∈ M is called an ε -chain in M if d ( x i − , x i ) < ε for all i ∈ { , . . . , n } . For any points x, y ∈ M let˘ d ε ( x, y ) = inf n n X i =1 d ( x i − , x i ) : x = x , . . . , x n = y is an ε -chain in M o ∪ (cid:8) ∞ (cid:9) . It is easy to see that the function ˘ d ε : M × M → [0 , ∞ ] has the following properties: • d ( x, y ) ≤ ˘ d ε ( x, y ); • ˘ d ε ( x, y ) = 0 if and only if x = y ; • ˘ d ε ( x, y ) = ˘ d ε ( y, x ); • ˘ d ε ( x, z ) ≤ ˘ d ε ( x, y ) + ˘ d ε ( y, z )for any points x, y, z ∈ M . In the last item we assume that r + ∞ = ∞ = ∞ + r for any r ∈ [0 , ∞ ].For any x, y ∈ M the finite or infinite number˘ d ( x, y ) = sup ε> ˘ d ε ( x, y )is called the intrinsic distance between the points x and y in the metric space M . If ˘ d ( x, y ) < ∞ for any points x, y ∈ M , then the intrinsic distance ˘ d is called the intrinsic metric of M .The properties of the distances ˘ d ε for ε > d : • d ( x, y ) ≤ ˘ d ( x, y ); • ˘ d ( x, y ) = 0 if and only if x = y ; SOMETRIES BETWEEN SMOOTH 2-DIMENSIONAL BANACH SPACES 5 • ˘ d ( x, y ) = ˘ d ( y, x ); • ˘ d ( x, z ) ≤ ˘ d ( x, y ) + ˘ d ( y, z );for any x, y, z ∈ M . Lemma 3.1. Let X be a smooth -dimensional Banach space and r : R → S X be a naturalparametrization of its unit sphere. For any closed interval [ a, b ] ⊂ R with r ([ a, b ]) = S X , therestriction r ′ ↾ [ a, b ] is an isometry of [ a, b ] onto the arc r ([ a, b ]) ⊂ S X endowed with its intrinsicmetric ˘ d .Proof. The condition k r ′ k = 1 implies that r : R → S X is a local homeomorphism. Since r ([ a, b ]) = S X , the restriction r ↾ [ a, b ] is injective. To see that it is an isometry, take any points x < y in thesegment [ a, b ]. Given any ε > ε -chain x = x < x < · · · < x n = y in [ a, b ]. Since themap r : R → S X is non-expanding (as k r ′ k = 1), the sequence r ( x ) = r ( x ) , . . . , r ( x n ) = r ( y ) isan ε -chain in r ( I ) with n X i =1 k r ( x i ) − r ( x i − ) k ≤ n X i =1 | x i − x i − | = | x − y | . Then ˘ d ε ( r ( x ) , r ( y )) ≤ | x − y | and hence ˘ d ( r ( x ) , r ( y )) ≤ | x − y | .We claim that ˘ d ( r ( x ) , r ( y )) = | x − y | . To derive a contradiction, assume that ˘ d ( r ( x ) , r ( y )) < | x − y | . Then x = y and ˘ d ( r ( x ) , r ( y )) > 0. Choose a number ε > ε − ε · ˘ d ( r ( x ) , r ( y )) < | x − y | . By the uniform continuity of the restriction r ′ ↾ [ a, b ], there exists δ > k r ′ ( t ) − r ′ ( s ) k < ε for any t, s ∈ [ a, b ] with | t − s | < δ .By the compactness of [ a, b ], the injective map r ↾ [ a, b ] is a topological embedding and by thecompactness of r ([ a, b ]), the inverse map ( r ↾ [ a, b ]) − : r ([ a, b ]) → [ a, b ] is uniformly continuous. So,there exists ǫ > | s − t | < δ for any s, t ∈ [ a, b ] with k r ( s ) − r ( t ) k < ǫ .By the definition of ˘ d ( r ( x ) , r ( y )) > 0, there exists a ǫ -chain r ( x ) = y , . . . , y n = r ( y ) in r ([ a, b ])such that P ni =1 k y i − − y i k < (1 + ε ) · ˘ d ( r ( x ) , r ( y )). For every i ∈ { , . . . , n } choose a (unique)point x i ∈ I such that y i = r ( x i ). The choice of ǫ ensures that | x i − − x i | < δ and hence k r ′ ( t ) − r ′ ( x i ) k < ε for every t ∈ [ x i − , x i ]. Observe that k r ( x i ) − r ( x i − ) k = (cid:13)(cid:13)(cid:13) Z x i x i − r ′ ( t ) dt (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) Z x i x i − r ′ ( x i ) dt + Z x i x i − ( r ′ ( t ) − r ′ ( x i )) dt (cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13) Z x i x i − r ′ ( x i ) dt (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13) Z x i x i − ( r ′ ( t ) − r ′ ( x i )) dt (cid:13)(cid:13)(cid:13) ≥ k r ′ ( x i ) k · | x i − x i − | − Z x i x i − k r ′ ( t ) − r ′ ( x i ) k dt ≥ · | x i − x i − | − ε · | x i − x i − | = (1 − ε ) · | x i − x i − | and hence(1+ ε ) · ˘ d ( r ( x ) , r ( y )) > n X i =1 k y i − y i − k = n X i =1 k r ( x i ) − r ( x i − ) k > (1 − ε ) n X i =1 | x i − x i − | ≥ (1 − ε ) | x − y | . Then | x − y | ≤ ε − ε · ˘ d ( r ( x ) , r ( y )) < | x − y | , which is a contradiction completing the proof of the equality | x − y | = ˘ d ( r ( x ) , r ( y ). (cid:3) Now we prove the main result of this section. Lemma 3.2. For any natural parametrizations r , r : R → S X of the unit sphere of a smooth -dimensional Banach space X , there exists an isometry Φ : R → R of the real line such that r = r ◦ Φ . TARAS BANAKH Proof. For every k ∈ { , } the differentiability of r k and the equality k r ′ k k = 1 imply that r k : R → S X is a covering map. By [5, 1.30], there exists a unique continuous maps Φ , Ψ : R → R such that r = r ◦ Φ and r = r ◦ Ψ. The uniqueness of liftings [5, 1.30] impliesthat Φ ◦ Ψ = Ψ ◦ Φ is the identity map of R , which means that Φ is a homeomorphism of thereal line. By the continuity of the map r , there exists an increasing sequence of real numbers( x n ) n ∈ Z such that r ([ x n , x n +1 ]) = S X for every n ∈ Z . For every n ∈ Z consider the realnumber y n = Φ( y n ) and observe that ( y n ) n ∈ Z is a monotone sequence of real numbers in the realline such that r ([ x n , x n +1 ]) = r ([ y n , y n +1 ]) for every n ∈ Z . By Lemma 3.1, for every n ∈ Z the maps r ↾ [ x n , x n +1 ] and r ↾ [ y n , y n +1 ] are isometries of the segments [ x n , x n +1 ] and [ y n , y n +1 ]onto the arc r ([ x n , x n +1 ] = r ([ y n , y n +1 ]) endowed with its intrinsic metric. Then the mapΦ ↾ [ x n , x n +1 ] = ( r ↾ [ y n , y n +1 ]) − ◦ ( r ↾ [ x n , x n +1 ]) is an isometry of the interval [ x n , x n +1 ] onto theinterval [ y n , y n +1 ]. Having this information, it is easy to conclude that the homeomorphism Φ isan isometry of the real line. (cid:3) The polar parametrization of the unit sphere of a -based Banach space By a 2 -based Banach space we understand a 2-dimensional real Banach space endowed with abasis.Let ( X, k · k ) be a 2-based Banach space and e , e be the basis of X . Let e ∗ , e ∗ : X → R bebiorthogonal functionals to the basis e , e , which means that e ∗ ( e ) = 1 = e ∗ ( e ) and e ∗ ( e ) = 0 = e ∗ ( e ) . On the Banach space X consider the equivalent (Euclidean) norm | · | defined by | x | = q | e ∗ ( x ) | + | e ∗ ( x ) | . The relation between the norms k · k and | · | is described by the constants c = min {k x k : x ∈ X, | x | = 1 } and C = max {k x k : x ∈ X, | x | = 1 } . For a real number t it will be convenient to denote the element cos( t ) e + sin( t ) e of X by e itX . Itis clear that | e itX | = 1. Definition 4.1. The map p : R → S X , p : t e itX k e itX k , is called the polar parametrization of the unit sphere S X = { x ∈ X : k x k = 1 } of the 2-basedBanach space X .The following lemma establishes the Lipschitz property of the polar parametrization. Lemma 4.2. For every t, ε ∈ R we have the lower and upper bounds: cC · | sin( ε ) | ≤ k p ( t + ε ) − p ( t ) k ≤ C c · | sin( ε ) | . Proof. Observe that k p ( t + ε ) − p ( t ) k = (cid:13)(cid:13)(cid:13) e i ( t + ε ) X k e i ( t + ε ) X k − e itX k e itX k (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) e i ( t + ε ) X k e itX k − e itX k e i ( t + ε ) X kk e i ( t + ε ) X k · k e itX k (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) e i ( t + ε ) X k e itX k − e itX k e itX k + e itX k e itX k − e itX k e i ( t + ε ) X kk e i ( t + ε ) X k · k e itX k (cid:13)(cid:13)(cid:13) ≤k e i ( t + ε ) X − e itX k · k e itX k + k e itX k · (cid:12)(cid:12) k e itX k − k e i ( t + ε ) X k (cid:12)(cid:12) c ≤ k e i ( t + ε ) X − e itX k · k e itX k c ≤ C | e i ( t + ε ) − e it | c = 2 C | e iε − | c = 4 C | sin( ε/ | c and k p ( t + ε ) − p ( t ) k ≥ c · | p ( t + ε ) − p ( t ) | = c · (cid:12)(cid:12)(cid:12) e i ( t + ε ) X k e i ( t + ε ) X k − e itX k e itX k (cid:12)(cid:12)(cid:12) ≥ c k e itX k min {| re i ( t + ε ) X − e itX | : r > } ≥ cC min {| re iε − | : r ≥ } = cC · | sin( ε ) | . (cid:3) Next we establish some differentiability properties of the polar parametrization. Lemma 4.3. The polar parametrization p : R → S X , p : R → e it k e it k , has the following properties: (1) p ( t + π ) = − p ( t ) for every t ∈ R ; (2) the function p has one-sided derivatives p ′− ( t ) = lim ε →− p ( t + ε ) − p ( t ) ε and p ′ + ( t ) = lim ε → +0 p ( t + ε ) − p ( t ) ε at each point t ∈ R ; (3) the set Λ p = { t ∈ R : p ′− ( t ) = p ′ + ( t ) } is at most countable; (4) the functions p ′− and p ′ + have bounded variation on bounded susbets of R ; (5) the function p is twice differentiable almost everywhere and its second derivative p ′′ ismeasurable and locally integrable; (6) cC ≤ min {k p ′− ( t ) k , k p ′ + ( t ) k} ≤ max {k p ′− ( t ) k , k p ′ + ( t ) k} ≤ C c for every t ∈ R .Proof. 1. Observe that for every t ∈ R p ( t + π ) = e i ( t + π ) k e i ( t + π ) k = − e it k − e it k = − e it k e it k = − p ( t ) . p locally, i.e., in a neighborhoodof any point ϕ ∈ R . Given a real number ϕ , consider the point p ( ϕ ) ∈ S X and chose a vector v ∈ X that is linearly independent with the vector p ( ϕ ). Observe that the function f : R → R , f : t 7→ k p ( ϕ ) + tv k is convex and strictly positive. By Theorem 3.7.4 in [7], the function f has one sided derivatives f ′− ( x ) and f ′ + ( x ) at each point x ∈ R . Moreover, the functions f ′− and f ′ + are non-decreasingand the set Λ f = { x ∈ R : f ′− ( x ) = f ′ + ( x ) } is at most countable. By Theorem 6.1.3 [7], thenon-decreasing functions f ′− and f ′ + have bounded variation on each compact subset of R . Thenthe function g : R → S X , g : t p ( ϕ ) + tv k p ( ϕ ) + tv k = p ( ϕ ) + tvf ( t )has two-sided derivatives g ′− ( t ) = v · f ( t ) − ( p ( ϕ ) + tv ) · f ′− ( t ) f ( t ) and g ′ + ( t ) = v · f ( t ) − ( p ( ϕ ) + tv ) · f ′ + ( t ) f ( t ) at each point t ∈ R . Moreover, the functions g ′− and g ′ + have bounded variation on boundedsubsets of R , see Theorem 6.1.9, 6.1.10, 6.1.11 in [7].Consider the unit sphere T = { e itX : t ∈ R } of the Banach space ( X, | · | ) and the diffeomorphism h : ( ϕ − π, ϕ + π ) → T \ (cid:8) − p ( ϕ ) | p ( ϕ ) | (cid:9) , h ( t ) e itX . Next, consider the diffeomorphic embedding e : R → T , e : t p ( ϕ )+ vt | p ( ϕ )+ vt | and observe that h − ◦ e : R → ( ϕ − π, ϕ + π ) is a diffeomorphism of R onto some open interval ( a, b ) ⊂ ( ϕ − π, ϕ + π ) that contains ϕ . Then δ = e − ◦ h : ( a, b ) → R is adiffeomorphism of ( a, b ) onto the real line. Observe that for any t ∈ ( a, b ) we have p ( t ) = g ◦ δ ( t ).Now the monononicity of the diffeomorphism δ and the properties of the function g imply that thefunction p has two-sided derivatives p ′− ( t ) = g ′− ( δ ( t )) · δ ′ ( t ) and p ′ + ( t ) = g ′ + ( δ ( t )) · δ ′ ( t ) at eachpoint t ∈ ( a, b ), the set { t ∈ ( a, b ) : p ′− ( t ) = p ′ + ( t ) } is at most countable and the functions p ′− and p ′ + have bounded variation on each compact subset of the interval ( a, b ), see Theorem 6.1.11 [7]. TARAS BANAKH 5. By Theorem 1.3.1 [7], each monotone function is differentiable almost everywhere and itsderivative is measurable and locally integrable. By Theorem 6.1.15 [7], each real function that hasbounded variation on bounded sets is the difference of two increasing functions. Consequently,any function that has bounded variation on a bounded interval is differentiable almost everywhereand its derivative is measurable and integrable. Since the functions p ′− and p ′ + coincide almosteverywhere and have bounded variation on bounded subsets of the real line, the function p is twicedifferentiable almost everywhere and its second derivative p ′′ is measurable and locally integrable.6. The upper and lower bounds for the norms of the one-sided derivatives p ′− ( t ) and p ′ + ( t ) canbe easily derived from Lemma 4.2. (cid:3) For two vectors x , y ∈ X we write x ⇈ y if x = α · y for some positive real number α . Lemma 4.4. Let t, u, α, β ∈ R be real numbers such that p ′ ( t ) ⇈ p ( u ) and p ′ ( u ) | p ′ ( u ) | = α · p ( u ) | p ( u ) | + β · p ( t ) | p ( t ) | . Then | α | ≤ C + cc ·| β | .Proof. In the Banach space X consider the basis b = e iuX = cos( u ) e + sin( u ) e = p ( u ) | p ( u ) | and b = e i ( u + π ) X = − sin( u ) e + cos( u ) e . Let b ∗ , b ∗ : X → R be the biorthogonal functionals to the basis b , b , which means that b ∗ ( b ) = 1 = b ∗ ( b ) and b ∗ ( b ) = 0 = b ∗ ( b ) . Let D = { z ∈ X : | z | = 1 } be the unit disk in the norm | · | . The definitions of the numbers c = min t ∈ R k e it X k and C = max t ∈ R k e it X k imply that C D ⊂ B X ⊂ c D . By the convexity of the unit B X , the tangent line p ( u ) + p ′ ( u ) · R to B X at the point p ( u ) ∈ B X ⊂ c D does not intersect the interior of the disk C D ⊂ B X , which implies that | b ∗ ( p ′ ( u )) || p ′ ( u ) | ≥ C · b ∗ ( p ( u )) = 1 C ·| p ( u ) | ≥ cC . Then cC ≤ | b ∗ ( p ′ ( u )) || p ′ ( u ) | = (cid:12)(cid:12) b ∗ ( p ′ ( u ) | p ′ ( u ) | ) (cid:12)(cid:12) = (cid:12)(cid:12) α · b ∗ (cid:0) p ( u ) | p ( u ) | (cid:1) + β · b ∗ (cid:0) p ( t ) | p ( t ) | (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) β · b ∗ (cid:0) p ( t ) | p ( t ) | (cid:1)(cid:12)(cid:12) ≤ | β | , which implies 1 ≤ Cc | β | . Finally, observe that the equality p ′ ( u ) | p ′ ( u ) | = α · p ( u ) | p ( u ) | + β · p ( t ) | p ( t ) | implies thedesired upper bound | α | ≤ | β | ≤ Cc ·| β | + | β | = C + cc · | β | . (cid:3) Lemma 4.5. Let t, u, α, β ∈ R be real numbers such that p ′ ( t ) ⇈ p ( u ) and p ′ ( u ) = α · p ( u ) + β · p ( t ) .Then | α | ≤ ( C + c ) Cc | β | .Proof. The equality p ′ ( u ) = α · p ( u )+ β · p ( t ) implies the equality p ′ ( u ) | p ′ ( u ) | = α | p ( u ) || p ′ ( u ) | p ( u ) | p ( u ) | + β | p ( t ) || p ′ ( u ) | p ( t ) | p ( t ) | .By Lemma 4.4, α | p ( u ) || p ′ ( u ) | ≤ C + cc · | β | · | p ( t ) || p ′ ( u ) | and hence | α | ≤ C + cc | p ( t ) || p ( u ) | | β | ≤ C + cc · Cc · | β | . (cid:3) The natural parametrization of the unit sphere of a -based Banach space Let ( X, k · k ) be a 2-based Banach space, e , e be its basis and p : R → S X , p : t e itX k e itX k = cos( t ) e + sin( t ) e k cos( t ) e + sin( t ) e k be the polar parametrization of the unit sphere S X = { x ∈ X : k x k = 1 } of X . By Lemmas 4.2and 4.3, the function p is Lipschitz, has one-sided derivatives p ′− and p ′ + and the set Λ p = { x ∈ SOMETRIES BETWEEN SMOOTH 2-DIMENSIONAL BANACH SPACES 9 R : p ′− ( t ) = p ′ + ( t ) } is at most countable. Since the derivatives p ′− , p ′ + have bounded norm, theyare integrable, so we can consider the continuous increasing function s : R → R , s : t Z t k p ′− ( t ) k dt = Z t k p ′ + ( t ) k dt. For t ∈ [0 , π ] the value s ( t ) can be thought as the length of the curve on the sphere S X betweenthe points p (0) and p ( t ) in the Banach space X (see Section 3 for a more precise formulation ofthis assertion).The number L = s ( π ) = Z π k p ′− ( t ) k dt = Z π k p ′ + ( t ) k dt is called the half-length of the sphere S X in X .The definition of the function s and Lemma 4.3 imply the following lemma describing thesmoothness properties of the function s . Lemma 5.1. The function s : R → R has the following properties: (1) s is an increasing Lipschitz function; (2) s has one-sided derivatives s ′− ( t ) = k p ′− ( t ) k and s ′ + ( t ) = k p ′ + ( t ) k at each point t ∈ R ; (3) the set Λ s = { t ∈ R : s ′− ( t ) = s ′ + ( t ) } is at most countable; (4) the functions s ′− and s ′ + have bounded variation on bounded subsets of R ; (5) the function s is twice differentiable almost everywhere and its second derivative s ′′ ismeasurable and locally integrable; (6) cC ≤ min { s ′− ( t ) , s ′ + ( t ) } ≤ max { s ′− ( t ) , s ′ + ( t ) } ≤ C c for every t ∈ R . It follows that the increasing continuous function s ( t ) has the inverse function t ( s ), which isalso increasing and continuous, and has the following properties that can be derived from thecorresponding properties of the function s . Lemma 5.2. The function t : R → R has the following properties: (1) t is an increasing Lipschitz function; (2) s has one-sided derivatives t ′− ( s ) = s ′− ( t ( s )) = k p ′− ( t ( s )) k and t ′ + ( s ) = s ′ + ( t ( s )) = k p ′ + ( t ( s )) k at each point s ∈ R ; (3) the set Λ t = { s ∈ R : t ′− ( s ) = t ′ + ( s ) } is at most countable and is equal to the set s (Λ s ) where Λ s = { t ∈ R : s ′− ( t ) = s ′ + ( t ) } ; (4) the functions t ′− = s ′− ◦ t and t ′ + = s ′ + ◦ t have bounded variation on bounded subsets of R ; (5) the function t is twice differentiable almost everywhere and its second derivative t ′′ ismeasurable and locally integrable; (6) c C ≤ min { t ′− ( s ) , t ′ + ( s ) } ≤ max { t ′− ( s ) , t ′ + ( s ) } ≤ Cc for every s ∈ R . Now we can introduce one of central notions in this paper. Definition 5.3. The function r : R → S X , r : s p ( t ( s )) , is called the natural parametrization of the sphere S X .We recall that the number L := s ( π ) = Z π k p ′− ( t ) k dt = Z π k p ′ + ( t ) k dt stands for the half-length of the sphere S X . Lemma 5.4. The natural parametrization r : R → S X of S X has the following properties: (1) r ( s + L ) = − r ( s ) for every s ∈ R ; (2) the function r has one sided derivatives r ′− ( s ) = lim ε →− r ( s + ε ) − r ( s ) ε and r ′ + ( s ) = lim ε → +0 r ( s + ε ) − r ( s ) ε at each point s ∈ R ; (3) the set Λ r = { s ∈ R : r ′− ( s ) = r ′ + ( s ) } is at most countable; (4) r is non-expanding and has k r ′− ( s ) k = k r ′ + ( s ) k = 1 for every s ∈ R ; (5) the functions r ′− and r ′ + have bounded variation on each compact subset of R ; (6) the function r is twice differentiable almost everywhere and its second derivative r ′′ ismeasurable and locally integrable. (7) If the Banach space X is AC -smooth, then the derivative r ′ is locally absolutely continu-ous.Proof. 1. By Lemma 4.3(1), p ( t + π ) = − p ( t ) for all t ∈ R , which implies p ′− ( t + π ) = − p ′− ( t )and k p ′− ( t + π ) k = k p ′− ( t ) k for all t ∈ R . By Lemma 2.3, R t + πt k p ′− ( u ) k du = R π k p ′− ( u ) k du = L for any t ∈ R , which implies s ( t + π ) = Z t + π k p ′− ( t ) k dt = Z t k p ′− ( t ) k dt + Z t + πt k p ′− ( t ) k dt = s ( t ) + L. Then for every s ∈ R we have s ( t ( s ) + π ) = s ( t ( s )) + L = s + L = s ( t ( s + L ))and t ( s + L ) = t ( s ) + π , by the injectivity of the function s . Consequently, r ( s + L ) = p ( t ( s + L )) = p ( t ( s ) + π ) = − p ( t ( s )) = − r ( s ) . r = p ◦ t and the correspondingproperties of the functions p and t , see Lemmas 4.3 and 5.2.7. If the Banach space X is AC -smooth, then its sphere admits a natural parametrization f : R → S X of S X whose derivative f ′ is locally absolutely continuous. By Lemma 3.2, there existsan isometry Φ of the real line such that r = f ◦ Φ. The local absolute continuity of the function f ′ implies the local absolute continuity of the function r ′ = ± f ′ ◦ Φ. (cid:3) Lemma 5.5. Let s, v, α, β ∈ R be real numbers such that r ′ ( s ) = r ( v ) and r ′ ( v ) = α · r ( v ) + β · r ( s ) .Then | α | ≤ ( C + c ) Cc | β | .Proof. Consider the real numbers t = t ( s ) and u = t ( v ) and observe that p ′ ( t ) = k p ′ ( t ) k · r ′ ( s ) = k p ′ ( t ) k · r ( v ) = k p ′ ( t ) k · p ( t ( v )) = k p ′ ( t ) k · p ( u ) ⇈ p ( u )and p ′ ( u ) = k p ′ ( u ) k · r ′ ( v ) = k p ′ ( u ) k· α · r ( v ) + k p ′ ( u ) k· β · r ( s ) = k p ′ ( u ) k· α · p ( u ) + k p ′ ( u ) k· β · p ( t ) . Now Lemma 4.5 ensures that k p ′ ( u ) k·| α | ≤ ( C + c ) Cc k p ′ ( u ) k·| β | . Taking into account that k p ′ ( u ) k ≥ cC > | α | ≤ ( C + c ) Cc | β | . (cid:3) The radial and tangential curvatures of a 2-based Banach space Let ( X, k·k ) be a 2-based Banach space. By Lemma 5.4, the natural parametrization r : R → S X of the unit sphere S X of X is twice differentiable almost everywhere and the second derivative r ′′ of r is measurable and locally integrable.Let ¨Ω r be the set of parameters s ∈ R at which the derivatives r ′ ( s ) and r ′′ ( s ) exist. It is easyto see that ¨Ω r is a Borel subset (of type G δσ ) in the real line. As we already know, the set ¨Ω r hasfull measure (which means that R \ ¨Ω r has Lebesgue measure zero). For every s ∈ ¨Ω r the norm k r ′′ ( s ) k of the vector r ′′ ( s ) is called the curvature of the sphere S X at the point r ( s ). Since thevectors r ( s ) and r ′ ( s ) form a basis of the linear space X , there are unique real numbers ρ ( s ) and τ ( s ) such that(1) r ′′ ( s ) = − ρ ( s ) · r ( s ) − τ ( s ) · r ′ ( s ) . The numbers ρ ( s ) and τ ( s ) are called the radial and tangential curvatures of the sphere S X atthe point r ( s ). The minus sign in the equation (1) is chosen to make the radial curvature ρ ( s )non-negative. SOMETRIES BETWEEN SMOOTH 2-DIMENSIONAL BANACH SPACES 11 The functions ρ and τ are measurable, being solutions of the system of two linear equationswith the measurable coefficients r , r ′ , r ′′ . To express ρ and τ via r , r ′ and r ′′ , denote by e ∗ , e ∗ thebiorthogonal functionals of the basic e , e of the Banach space X . By the Cramer’s rule,(2) ρ = ( e ∗ ◦ r ′′ ) · ( e ∗ ◦ r ′ ) − ( e ∗ ◦ r ′′ ) · ( e ∗ ◦ r ′ )( e ∗ ◦ r ) · ( e ∗ ◦ r ′ ) − ( e ∗ ◦ r ) · ( e ∗ ◦ r ′ ) τ = ( e ∗ ◦ r ) · ( e ∗ ◦ r ′′ ) − ( e ∗ ◦ r ) · ( e ∗ ◦ r ′′ )( e ∗ ◦ r ) · ( e ∗ ◦ r ′ ) − ( e ∗ ◦ r ) · ( e ∗ ◦ r ′ ) . These formulas imply that the functions ρ and τ inherit many properties from the functions r , r ′ , r ′′ . In particular, the periodicity of the vector-function r , r ′ , r ′′ implies the periodicity of thefunctions ρ and τ . Lemma 6.1. ρ ( s + L ) = ρ ( s ) and τ ( s + L ) = τ ( s ) for all s ∈ ¨Ω r .Proof. Lemma 5.4(1) implies that r ( s + L ) = − r ( s ) , r ′ ( s + L ) = − r ′ ( s ) , and r ′′ ( s + L ) = − r ′′ ( s )and hence ρ ( s + L ) · r ( s + L ) + τ ( s + L ) · r ′ ( s + L ) = − r ′′ ( s + L ) = r ′′ ( s ) = − ρ ( s ) · r ( s ) − τ ( s ) · r ′ ( s ) = ρ ( s ) · r ( s + L ) + τ ( s ) · r ′ ( s + L ) . The linear independence of the vectors r ( s + L ) and r ′ ( s + L ) implies that ρ ( s + L ) = ρ ( s ) and τ ( s + L ) = τ ( s ). (cid:3) The radial and tangential curvatures determine an absolutely smooth 2-based Banach spaceuniquely up to a linear isometry. Theorem 6.2. Two AC -smooth -dimensional Banach spaces Z, Y are linearly isometric if andonly if for some bases in Z, Y the corresponding radial and tangential curvatures of the -basedBanach spaces Z, Y coincide.Proof. The “only if” part is trivial and holds for any (not necessarily absolutely smooth) Banachspaces Y, Z . To prove the “if” part, assume that the Banach spaces Y, Z have bases such that ρ Y = ρ Z and τ Y = τ Z , where ρ Y and τ Y (resp. ρ Z and τ Z ) are the radial and tangential curvaturesof the unit sphere of the 2-based Banach space Y (resp. Z ).Let r Y and r Z be the natural parametrizations of the spheres S Y and S Z of the 2-based Banachspaces Y, Z , respectively. By Lemma 3.2, the AC -smoothness of the Banach spaces Y, Z impliesthe local absolute continuity of the derivatives r ′ Y and r ′ Z . Then the second derivatives r ′′ Y and r ′′ Z exist almost everywhere and for every s ≥ Z s r ′′ Y ( t ) dt = r ′ Y ( s ) − r ′ Y (0) and Z s r ′′ Z ( t ) dt = r ′ Z ( s ) − r ′ Z (0) , see Theorem 7.1.5 in [7].By the Cramer’s formulas (2), the continuity of the functions r Y and r ′ Y and the local integra-bility of the function r ′′ Y imply the local integrability of the functions ρ Y and τ Y .Let F : Y → Z be a linear operator such that F ( r Y (0)) = r Z (0) and F ( r ′ Y (0)) = r ′ Z (0). Weclaim that F ◦ r Y ( s ) = r Z ( s ) for all s ≥ 0. It suffices to show that the set E = { s ∈ R : F ◦ r Y ( s ) = r Z ( s ) and F ◦ r ′ Y ( s ) = r ′ Z ( s ) } contains the half-line [0 , ∞ ). Let a = sup { s ∈ [0 , ∞ ) : [0 , s ] ⊂ E } . If a = ∞ , then [0 , ∞ ) ⊂ E andwe are done. So, assume that a is finite.By the local integrability of the functions ρ := ρ Y = ρ Z and τ := τ Y = τ Z , there exists apositive real number ε < Z a + εa ( | ρ ( t ) | + | τ ( t ) | ) dt < . Let M = max t ∈ [ a,a + ε ] max {k F ( r Y ( t )) − r Z ( t ) k Z , k F ( r ′ Y ( t )) − r ′ Z ( t ) k Z } and δ ∈ [0 , ε ] be a real number such thatmax {k F ( r Y ( a + δ )) − r Z ( a + δ ) k Z , k F ( r ′ Y ( a + δ )) − r ′ Z ( a + δ ) k Z } = M. The definition of the number a implies that M > k F ◦ r Y ( a + δ ) − r Z ( a + δ ) k Z = k F ◦ r Y ( a + δ ) − F ◦ r Y ( a ) + r Z ( a ) − r Z ( a + δ ) k Z = (cid:13)(cid:13) Z a + δa ( F ( r ′ Y ( t )) − r ′ Z ( t )) dt (cid:13)(cid:13) Z ≤ Z a + δa k F ( r ′ Y ( t )) − r ′ Z ( t ) k Z dt ≤ M · δ < M, and k F ◦ r ′ Y ( a + δ ) − r ′ Z ( a + δ ) k Z = (cid:13)(cid:13) Z a + δa ( F ( r ′′ Y ( t )) − r ′′ Z ( t )) dt (cid:13)(cid:13) Z ≤ Z a + δa k F ( r ′′ Y ( t )) − r ′′ Z ( t ) k dt = Z a + δa k − F ( ρ Y ( t ) r Y ( t ) + τ Y ( t ) r ′ Y ( t )) + ( ρ Z ( t ) r Z ( t ) + τ Z ( t ) r ′ Z ( t )) k Z dt = Z a + δa k ρ ( t )( r Z ( t ) − F ( r Y ( t ))) + τ ( t )( r ′ Z ( t )) − F ( r ′ Y ( t )) k Z dt ≤ Z a + δa (cid:0) | ρ ( t ) | · k F ( r Y ( t )) − r Z ( t ) k Z + | τ ( t ) | · k F ( r ′ Y ( t )) − r ′ Z ( t ) k Z (cid:1) dt ≤ M · Z a + δa ( | ρ ( t ) | + | τ ( t ) | ) dt < M, which contradicts the definition of the number M . This contradiction shows that a = ∞ . Then F ( S Y ) = { F ( r Y ( s )) : s ≥ } = { r Z ( s ) : s ≥ } = S Z , which implies that F is a linear isometry of the Banach spaces X and Y . (cid:3) Example 6.3. For the complex plane C considered as a 2-based Banach space with basic vectors e = 1 and e = i and norm k z k = | z | we have r ( s ) = p ( s ) = e is , r ′ ( s ) = ie is , r ′′ ( s ) = − e is = − r ( s )and hence ρ ( s ) = 1 and τ ( s ) = 0 for all s ∈ R . Proposition 6.4. For a -based Banach space X the following conditions are equivalent: (1) X is isometric to a Hilbert space; (2) X has radial curvature ρ ( s ) = 1 and tangential curvature τ ( s ) = 0 for any s ∈ R . (3) X is smooth and has constant radial and tangential curvatures.Proof. (1) ⇒ (2) If X is isometric to a Hilbert space, then it is linearly isometric to the Hilbertspace ( C , | · | ). By Example 6.3, ρ ( s ) = 1 and τ ( s ) = 0 for all s ∈ R .(2) ⇒ (1) If ρ ( s ) = 1 and τ ( s ) = 0 for all s ∈ R , then r ′′ ( s ) = − r ( s ) is continuous. Then thederivative r ′ is continuously differentiable and hence locally absolutely continuous. By (the proofof) Theorem 6.2, the Banach space X is linearly isometric to the Hilbert space ( C , | · | ).(1) ⇒ (3) The implication (1) ⇒ (3) follows from Example 6.3.(3) ⇒ (1) Assume that X is smooth and has constant radial and tangential curvatures ρ and τ . Then the natural parametrization r of X satisfies the differential equation r ′′ + τ r ′ + ρ r = 0with constant coefficients, which implies that r ′ is continuously differentiable and hence locallyabsolutely continuous. Solving this differential equation r ′′ + τ r ′ + ρ r = 0 by the standard method[15, § x + τ x + ρ = 0 are nonzero but have zero real parts. This happens only if τ = 0 and ρ > 0. In this case the natural parametrization r is of the form r ( s ) = x e i √ ρs + x e − i √ ρs for SOMETRIES BETWEEN SMOOTH 2-DIMENSIONAL BANACH SPACES 13 some linearly independent vectors x , x ∈ X . Now we see that the unit sphere of X coincideswith the ellipse { x e i √ ρs + x e − i √ ρs : s ∈ R } and X is isometric to a Hilbert space. (cid:3) Remark 6.5. Any polyhedral 2-based Banach space X has radial and tangential curvaturesequal to zero everywhere except for finitely many points. Using the Cantor function (which isnon-decreasing and non-constant, but has zero derivative almost everywhere, see [7, 1.3.3]), itis possible to construct two non-isometric smooth two-dimensional Banach spaces whose radialand tangential curvatures are equal to zero almost everywhere. This shows that the absolutesmoothness of the Banach spaces Y, Z in Theorem 6.2 cannot be weakened to the smoothness of Y and Z . Remark 6.6. Theorem 6.2 is actually the standard Uniqueness Theorem for solutions of thesecond order linear differential equation(3) r ′′ + τ · r ′ + ρ · r = 0in a suitable Sobolev space (see [8]). Observe that the natural parametrization r of the unit sphere S X of a 2-based Banach space X is 2 L -periodic and the radial and tangential curvatures ρ and τ are L -periodic functions. The problem of the existence of a periodic solution of the differentialequation (3) with periodic coefficients ρ and τ is a subject of the Floquet Theory [4] (see also [9]),which has many applications, in particular, to Celestial Mechanics. One of conclusions of thistheory will be exploited in the proof of Lemma 8.7. Exercise 6.7. Calculate the (radial and tangential) curvature of the unit sphere of the 2-basedBanach space ( C , k · k ) endowed with the basic vectors 1 and i and norm k x + iy k = ( | x | p + | y | p ) p for some p ∈ (1 , ∞ ).7. Radial and tangential supercurvatures of a Banach space In this section for any smooth 2-based Banach space X we introduce two functions P, T : R → R which are tightly related to the radial and tangential curvatures of X but have better continuityand differentiability properties.If the 2-based Banach space X is smooth, then its natural parametrization r is continuouslydifferentiable and for every s ∈ R the vector r ′ ( s ) is well-defined and belongs to the unit sphere S X .Then we can find a unique real number ϕ ( s ) in the interval ( s, s + 2 L ) such that r ′ ( s ) = r ( ϕ ( s ).It can be shown that the function ϕ : R → R is continuous and non-decreasing. Moreover, theBanach space X is strictly convex if and only if the function ϕ is strictly increasing. The function ϕ will be called the phase shift of the natural parametrization r .Observe that for every s ∈ R the vectors r ( s ) and r ′ ( s ) form a basis of the linear space X .Consequently, we can write the vector r ′ ( ϕ ( s )) as the linear combination r ′ ( ϕ ( s )) = P ( s ) · r ( s ) + T ( s ) · r ′ ( s )for unique real numbers P ( s ) and T ( s ), called the radial and tangential supercurvatures of thesphere S X of X .In the following lemma we establish some properties of the radial and tangential supercurva-tures. Lemma 7.1. If the -based Banach space X is smooth, then (1) P and T are continuous functions such that | T ( s ) | ≤ ( C + c ) Cc | P ( s ) | and | P ( s ) | ≥ c C + Cc + c for all s ∈ R . (2) If X is AC -smooth, then the functions ϕ , P , T and TP are locally absolutely continuous. (3) For any s ∈ ¨Ω r we have k r ′′ ( s ) k = ϕ ′ ( s ) , ρ ( s ) = P ( s ) · ϕ ′ ( s ) , τ ( s ) = T ( s ) · ϕ ′ ( s ) , and | τ ( s ) | ≤ ( C + c ) Cc | ρ ( s ) | .Proof. 1. If the Banach space X is smooth, then the map r ′ = r ◦ ϕ is continuous and by theCramer’s formulas (resembling (2)), the functions P and T can be expresses via the continuousfunctions r ′ ◦ ϕ , r and r ′ and hence are continuous. Since r ′ ( s ) = r ( ϕ ( s )) and r ′ ( ϕ ( s )) = P ( s ) · r ( s ) + T ( s ) · r ′ ( s ) = T ( s ) · r ( ϕ ( s )) + P ( s ) · r ( s ) , we can apply Lemma 5.5 to conclude that | T ( s ) | ≤ ( C + c ) Cc | P ( s ) | . Then1 = k r ′ ( ϕ ( s )) k ≤ | P ( s ) k·k r ( s ) k + | T ( s ) |·k r ′ ( s ) k ≤ | P ( s ) | + ( C + c ) Cc | P ( s ) | = C + Cc + c c ·| P ( s ) | , which implies | P ( s ) | ≥ c C + Cc + c .2. If X is AC -smooth, then by Lemma 5.4, the function r ′ = r ◦ ϕ is locally absolutelycontinuous. Using Lemma 3.1, it is possible to prove that the local absolute continuity of thefunction r ◦ ϕ implies the local absolute continuity of the function ϕ . By the monotonicity of ϕ , the composition r ′ ◦ ϕ is locally absolutely continuous. The functions P and T are locallyabsolutely continuous since they can be expressed by the Cramer formula via the locally absolutelycomtinuous functions r ′ ◦ ϕ , r and r ′ (see also Theorem 7.1.10 [7] for the preservation of absolutelycontinuous functions by algebraic operations). Since | P | ≥ c C + Cc + c > 0, the function TP iswell-defined and locally absolutely continuous.3. For any s ∈ ¨Ω r we get r ′ ( s ) = r ( ϕ ( s )) and hence r ′′ ( s ) = r ′ ( ϕ ( s )) · ϕ ′ ( s ) and k r ′′ ( s ) k = k r ′ ( ϕ ( s ) k · | ϕ ′ ( s ) | = ϕ ′ ( s ). The linear independence of the vectors r ( s ) , r ′ ( s ) and the equality ρ ( s ) r ( s ) + τ ( s ) r ′ ( s ) = − r ′′ ( s ) = − r ′ ( ϕ ( s )) · ϕ ′ ( s ) = (cid:0) P ( s ) r ( s ) + T ( s ) r ′ ( s ) (cid:1) · ϕ ′ ( s )imply the equalities ρ ( s ) = P ( s ) ϕ ′ ( s ) and τ ( s ) = T ( s ) · ϕ ′ ( s ). Then τ ( s ) = T ( s ) · ϕ ′ ( s ) = T ( s ) P ( s ) P ( s ) · ϕ ′ ( s ) = T ( s ) P ( s ) ρ ( s ) and hence | τ ( s ) | = | T ( s ) || P ( s ) | · | ρ ( s ) | ≤ ( C + c ) Cc · | ρ ( s ) | . (cid:3) Lemma 7.1 implies that for an ( AC -)smooth 2-based Banach space X , the function ψ = TP is well-defined and (locally absolutely) continuous. Moreover, if s ∈ ¨Ω r and ρ ( s ) > 0, then τ ( s ) ρ ( s ) = ψ ( s ). The function ψ = TP will be called the quotient curvature of the shere S X . Thisfunction will play a crucial role is calculating the tangential curvature τ using measurements ofdistances on the sphere S X .8. Calculating the radial and tangential curvatures In this section we derive formulas for calculating the radial and tangential curvatures via themetric of the sphere S X of a 2-based Banach space ( X, k · k ).Denote by ¨Ω r the set of parameters s ∈ R at which the second derivative r ′′ ( s ) of the naturalparametrization r exists. Lemma 8.1. For any s ∈ ¨Ω r we have the equality ρ ( s ) = lim ε → − k r ( s + ε ) − r ( s + L − ε ) k ε ≥ . Proof. By the Taylor formula, for a small ε we have the expansion r ( s + ε ) = r ( s ) + r ′ ( s ) ε + r ′′ ( s ) ε + o ( ε )and hence r ( s + ε ) + r ( s − ε ) = 2 r ( s ) + r ′′ ( s ) ε + o ( ε ) . Taking into account that r ( s + L ) = − r ( s ), we can observe that r ( s + ε ) − r ( s + L − ε ) = r ( s + ε ) + r ( s − ε ) = 2 r ( s ) + r ′′ ( s ) ε + o ( ε ) =2 r ( s ) − ( ρ ( s ) r ( s ) + τ ( s ) r ′ ( s )) ε + o ( ε ) = (2 − ρ ( s ) ε ) r ( s ) − τ ( s ) ε r ′ ( s ) + o ( ε ) =(2 − ρ ( s ) ε ) (cid:0) r ( s ) − τ ( s ) ε − ρ ( s ) ε r ′ ( s ) + o ( ε ) (cid:1) = (2 − ρ ( s ) ε ) (cid:0) r ( s ) − τ ( s ) ε r ′ ( s ) + o ( ε ) (cid:1) =(2 − ρ ( s ) ε ) (cid:0) r ( s − τ ( s ) ε ) + o ( ε ) (cid:1) Then k r ( s + ε ) − r ( s + L − ε ) k = (2 − ρ ( s ) ε ) k r ( s − τ ( s ) ε ) + o ( ε ) k = 2 − ρ ( s ) ε + o ( ε ) SOMETRIES BETWEEN SMOOTH 2-DIMENSIONAL BANACH SPACES 15 and hence ρ ( s ) = lim ε → − k r ( s + ε ) − r ( s + L − ε ) k ε ≥ . (cid:3) Calculating the tangential curvature is much more tricky and exploits the following series oflemmas. Lemma 8.2. Assume that the Banach space X is AC -smooth and a point s ∈ R satisfied thefollowing conditions: (1) s is a Lebesgue point of the function r ′′ ; (2) ρ ( s ) > ; (3) the quotient curvature ψ = TP = τρ is differentiable at s .Then for small ε we have the asymptotic formula k r ( s + ε ) − r ( s − ε ) k = 2 ε − ρ ( s )( ψ ′ ( s ) + 1) ε + o ( ε ) which allows us to find the derivative of the quotient curvature ψ at s : ψ ′ ( s ) = − ε → ε − k r ( s + ε ) − r ( s − ε ) k ρ ( s ) · ε . Proof. By Lemma 3.2, the natural parametrization r has locally absolutely continuous derivative r ′ , which allows us to write r ( s + ε ) − r ( s − ε ) = Z ε − ε r ′ ( s + t ) dt = 2 r ′ ( s ) ε + Z ε − ε ( r ′ ( s + t ) − r ′ ( s )) dt =2 r ′ ( s ) ε + Z ε − ε Z t r ′′ ( s + u ) du dt = 2 r ′ ( s ) ε + Z ε Z t (cid:0) r ′′ ( s + u ) − r ′′ ( s − u ) (cid:1) du dt. Since s is a Lebesgue point of the function r ′′ , the Cramer’s formulas (2) imply that s is aLebesgue point of the locally integrable functions ρ and τ . Then the functions ρ ( s + u ) − ρ ( s − u )and τ ( s + u ) − τ ( s − u ) depending on the small positive parameter u are of order o [1].Now using the formula r ′′ = − ρ r − τ r ′ , we establish the asymptotics of the function r ′′ ( s + u ) − r ′′ ( s − u ) up to order o [ u ]: r ′′ ( s + u ) − r ′′ ( s − u ) = − ρ ( s + u ) r ( s + u ) − τ ( s + u ) r ′ ( s + u ) + ρ ( s − u ) r ( s − u ) + τ ( s − u ) r ′ ( s − u ) =( ρ ( s − u ) − ρ ( s + u )) r ( s + u ) + ρ ( s − u )( r ( s − u ) − r ( s + u ))+( τ ( s − u ) − τ ( s + u )) r ′ ( s + u ) + τ ( s − u )( r ′ ( s − u ) − r ′ ( s + u )) =( ρ ( s − u ) − ρ ( s + u ))( r ( s ) + r ′ ( s ) u + o ( u )) − ρ ( s − u )( r ′ ( s ) u + o ( u ))+( τ ( s − u ) − τ ( s + u ))( r ′ ( s ) + r ′′ ( s ) u + o ( u )) − τ ( s − u )( r ′′ ( s ) u + o ( u )) =( ρ ( s − u ) − ρ ( s + u )) r ( s ) + o [ u ] − ρ ( s − u ) − ρ ( s ))( r ′ ( s ) u + o ( u )) − ρ ( s )( r ′ ( s ) u + o ( u ))+( τ ( s − u ) − τ ( s + u )) r ′ ( s ) + o [ u ] − τ ( s − u ) − τ ( s ))( r ′′ ( s ) u + o ( u )) − τ ( s )( r ′′ ( s ) u + o ( u )) =( ρ ( s − u ) − ρ ( s + u )) r ( s ) + o [ u ] − ρ ( s ) r ′ ( s ) u +( τ ( s − u ) − τ ( s + u )) r ′ ( s ) + o [ u ] + 2 τ ( s )( ρ ( s ) r ( s ) + τ ( s ) r ′ ( s )) u = r ( s )( ρ ( s − u ) − ρ ( s + u ) + 2 τ ( s ) ρ ( s ) u + o [ u ]) + r ′ ( s )( τ ( s − u ) − τ ( s + u ) − ρ ( s ) u + 2 τ ( s ) u + o [ u ]) . After integrating and using Lemma 2.2, we obtain r ( s + ε ) − r ( s − ε ) = 2 r ′ ( s ) ε + r ( s ) Z ε Z t (cid:0) ρ ( s − u ) − ρ ( s + u ) + 2 τ ( s ) ρ ( s ) u + o [ u ] (cid:1) du dt + r ′ ( s ) Z ε Z t (cid:0) τ ( s − u ) − τ ( s + u ) − ρ ( s ) u + 2 τ ( s ) u + o [ u ] (cid:1) du dt = r ( s ) (cid:16) τ ( s ) ρ ( s ) ε + o ( ε ) + Z ε Z t ( ρ ( s − u ) − ρ ( s + u )) du dt (cid:17) + r ′ ( s ) (cid:16) ε + ( τ ( s ) − ρ ( s )) ε + o ( ε ) + Z ε Z t (cid:0) τ ( s − u ) − τ ( s + u ) (cid:1) du dt (cid:17) . Taking into account that s is a Lebesgue point of the functions ρ and τ , we can prove that thedouble integrals Z ε Z t (cid:0) ρ ( s − u ) − ρ ( s + u ) (cid:1) du dt and Z ε Z t (cid:0) τ ( s − u ) − τ ( s + u ) (cid:1) du dt are of order o ( ε ).Observe that for a small δ , by the differentiability of r ′ at s , we have the development r ′ ( s + δ ) = r ′ ( s ) + r ′′ ( s ) δ + o ( δ ) = (cid:0) − ρ ( s ) δ + o ( δ ) (cid:1) r ( s ) + (cid:0) − τ ( s ) δ + o ( δ ) (cid:1) r ′ ( s ) . Since the function r is differentiable with k r ′ k ≡ 1, the map r : R → S X is a local homeomorphism.Consequently, for a small ε there exists a small δ ( ε ) = o (1) such that r ( s + ε ) − r ( s − ε ) = k r ( s + ε ) − r ( s − ε ) k · r ′ ( s + δ ( ε )) . Our task is to develop the function n ( ε ) = k r ( s + ε ) − r ( s − ε ) k into a power series.Let f ∗ , f ∗ : X → R be the linear functionals, which are biorthogonal to the basis r ( s ) and r ′ ( s ).This means that f ∗ ( r ( s )) = 1 = f ∗ ( r ′ ( s )) and f ∗ ( r ′ ( s )) = 0 = f ∗ ( r ( s )) . To develop the functions n ( ε ) and δ ( ε ) into power series, we shall use the equalities f ∗ i ( r ( s + ε ) − r ( s − ε )) = n ( ε ) · f ∗ i ( r ′ ( s + δ ( ε )) , i ∈ { , } , one after the other, increasing the precision of the development.We start with the equality2 ε (1 + o (1)) = f ∗ ( r ( s + ε ) − r ( s )) = n ( ε ) · f ∗ ( r ′ ( s ) + o (1))implying n ( ε ) = 2 ε + o ( ε ) and hence n ( ε ) = 2 ε + n ( ε ) for some function n ( ε ) = o ( ε ).The equality τ ( s ) ρ ( s ) ε + o ( ε ) + Z ε Z t ( ρ ( s − u ) − ρ ( s + u )) du dt = f ∗ ( r ( s + ε ) − r ( s − ε )) = n ( ε ) · f ∗ ( r ′ ( s + δ ( ε ))) = 2 ε (1 + o (1))( − ρ ( s ) δ ( ε ))(1 + o (1))implies δ ( ε ) = − τ ( s ) ε − ρ ( s ) ε Z ε Z t ( ρ ( s − u ) − ρ ( s + u )) du dt + o ( ε ) . SOMETRIES BETWEEN SMOOTH 2-DIMENSIONAL BANACH SPACES 17 Finally, the equality2 ε + ( τ ( s ) − ρ ( s )) ε + o ( ε ) + Z ε Z t (cid:0) τ ( s − u ) − τ ( s + u ) (cid:1) du dt (2 ε + n ( ε ))(1 − τ ( s ) δ ( ε ) + o ( δ ( ε )) =(2 ε + n ( ε )) (cid:0) τ ( s ) ε + τ ( s )2 ρ ( s ) ε Z ε Z t ( ρ ( s − u ) − ρ ( s + u )) du dt + o ( ε ) (cid:1) =2 ε + τ ( s ) ε + τ ( s ) ρ ( s ) Z ε Z t ( ρ ( s − u ) − ρ ( s + u )) du dt + n ( ε ) (cid:16) τ ( s )2 ρ ( s ) ε Z ε Z t ( ρ ( s − u ) − ρ ( s + u )) du dt (cid:17) + o ( ε ) =2 ε + τ ( s ) ε + τ ( s ) ρ ( s ) Z ε Z t ( ρ ( s − u ) − ρ ( s + u )) du dt + n ( ε )(1 + o (1)) + o ( ε )implies n ( ε ) = (1+ o (1)) (cid:16) − ρ ( s ) ε + o ( ε ) + Z ε Z t (cid:0) τ ( s − u ) − τ ( s + u ) − τ ( s ) ρ ( s ) ( ρ ( s − u ) − ρ ( s + u )) (cid:1) du dt (cid:17) =(1+ o (1)) (cid:16) − ρ ( s ) ε + o ( ε ) − ρ ( s ) Z ε Z t ρ ( s )( τ ( s + u ) − τ ( s − u )) − τ ( s )( ρ ( s + u ) − ρ ( s − u )) ρ ( s ) du dt (cid:17) . To calculate the latter double integral, let us observe that ρ ( s )( τ ( s + u ) − τ ( s − u )) − τ ( s )( ρ ( s + u ) − ρ ( s − u )) ρ ( s ) = ρ ( s ) τ ( s + u ) − τ ( s ) ρ ( s + u ) ρ ( s ) + τ ( s ) ρ ( s − u ) − τ ( s − u ) ρ ( s ) ρ ( s ) = ρ ( s + u ) ρ ( s ) · ρ ( s ) τ ( s + u ) − τ ( s ) ρ ( s + u ) ρ ( s ) ρ ( s + u ) + ρ ( s − u ) ρ ( s ) · τ ( s ) ρ ( s − u ) − τ ( s − u ) ρ ( s ) ρ ( s ) ρ ( s − u ) = ρ ( s + u ) ρ ( s ) ( ψ ( s + u ) − ψ ( s )) + ρ ( s − u ) ρ ( s ) ( ψ ( s ) − ψ ( s − u )) =(1 + o [1])( ψ ′ ( s ) u + o ( u )) + (1 + o [1])( ψ ′ ( s ) u + o ( u )) = 2 ψ ′ ( s ) u + o [ u ] . Applying Lemma 2.2, we eventually obtain n ( ε ) = 2 ε + n ( ε ) = 2 ε + (1+ o (1)) (cid:16) − ρ ( s ) ε + o ( ε ) − ρ ( s ) Z ε Z t (2 ψ ′ ( s ) u + o [ u ]) du dt (cid:17) =2 ε − ρ ( s )(1 + ψ ′ ( s )) ε + o ( ε ) . (cid:3) Lemma 8.3. If the Banach space X is A ˘ C -smooth, then for almost every s ∈ R the derivative ψ ′ of the quotient curvature ψ = TP can be calculated by the formula ψ ′ ( s ) = − ε → ε − k r ( s + ε ) − r ( s − ε ) k ρ ( s ) · ε . Proof. By Lemma 5.4, the AC -smoothness of X implies the local absolute continuity of thefunctions r ′ . By Lemma 7.1, the functions P and T are locally absolutely continuous and hencethey are differentiable at points of the sets ˙Ω P and ˙Ω T of full measure, respectively. By Lemma 5.4,the function r ′′ is locally integrable and by the Lebesgue Theorem 7.1.21 in [7], the set Ω r ′′ ofLebesgue points of the locally integrable function r ′′ has full measure. By Theorem 7.1.20 [7],each point s ∈ Ω r ′′ is belongs to the set ¨Ω r of twice differentiability of r . The A ˘ C -smoothnessof X and Lemma 3.2 imply that the set Ω ρ := { s ∈ ¨Ω r : ρ ( s ) > } has full measure. Then theset Ω = Ω ρ ∩ Ω r ′′ ∩ Ω P ∩ Ω T has a full measure and we can apply Lemma 8.2 to see that for any s ∈ Ω the derivative ψ ′ ( s ) can be calculated by the formula given in Lemma 8.3. (cid:3) Next we consider the case of AC L -smooth Banach spaces. Lemma 8.4. Assume that a point s ∈ R satisfied the following conditions: (1) s ∈ ¨Ω r ; (2) the function r ′ is Lipschitz at ϕ ( s ) ; (3) the functions P and T are differentiable at s ; (4) ρ ( s ) = 0 .Then ψ ′ ( s ) = − .Proof. Since s ∈ ¨Ω r , the numbers ρ ( s ) and τ ( s ) are well-defined. Taking into account that r ′ ( s ) = r ( ϕ ( s )), we can prove that ϕ is differentiable at s and r ′′ ( s ) = r ′ ( ϕ ( s )) · ϕ ′ ( s ), whichimplies ϕ ′ ( s ) = k r ′′ ( s ) k .By Lemma 7.1, | τ ( s ) | ≤ ( C + c ) Cc ρ ( s ). Then the equality ρ ( s ) = 0 implies τ ( s ) = 0, r ′′ ( s ) = − ρ ( s ) r ( s ) − τ ( s ) r ′ ( s ) = 0, and ϕ ′ ( s ) = k r ′′ ( s ) k = 0.By the Lipschitz property of r ′ at ϕ ( s ), we have r ′ ( ϕ ( s + ε )) = r ′ (cid:0) ϕ ( s ) + ϕ ′ ( s ) + o ( ε ) (cid:1) = r ′ (cid:0) ϕ ( s ) ε + o ( ε ) (cid:1) = r ′ ( ϕ ( s )) + o ( ε ) . On the other hand, P ( s ) r ( s ) + T ( s ) r ′ ( s ) = − r ′ ( ϕ ( s )) = − r ′ ( ϕ ( s + ε )) + o ( ε ) = P ( s + ε ) r ( s + ε ) + T ( s + ε ) r ′ ( s + ε ) + o ( ε ) =( P ( s ) + P ′ ( s ) ε + o ( ε ))( r ( s ) + r ′ ( s ) ε + o ( ε )) + ( T ( s ) + T ′ ( s ) ε + o ( ε ))( r ′ ( s ) + o ( ε )) =( P ( s ) + P ′ ( s ) ε + o ( ε )) r ( s ) + ( T ( s ) + ( P ( s ) + T ′ ( s )) ε + o ( ε )) r ′ ( s ) , which implies P ′ ( s ) = 0 and P ( s ) + T ′ ( s ) = 0. Then ψ ′ ( s ) = (cid:16) TP (cid:17) ′ ( s ) = T ′ ( s ) P ( s ) − T ( s ) P ′ ( s ) P ( s ) = − . (cid:3) Lemma 8.5. If the Banach space X is strictly convex and AC L -smooth, then for almost every s ∈ R the derivative ψ ′ ( s ) of the quotient curvature ψ = TP can be calculated by the formula ψ ′ ( s ) = − ε → ε − k r ( s + ε ) − r ( s − ε ) k ρ ( s ) · ε if ρ ( s ) > − if ρ ( s ) = 0 . Proof. By Lemma 5.4, the AC -smoothness of X implies the local absolute continuity of thefunctions r ′ . By Lemma 7.1, the functions P and T are locally absolutely continuous and hencethey are differentiable at points of the sets ˙Ω P and ˙Ω T of full measure, respectively. By Lemma 5.4,the function r ′′ is locally integrable and by the Lebesgue Theorem 7.1.21 in [7], the set Ω r ′′ ofLebesgue points of the locally integrable function r ′′ has full measure. By Theorem 7.1.20 [7],each point s ∈ Ω r ′′ is belongs to the set ¨Ω r of twice differentiability of r . By the AC L -smoothnessof X and Lemma 3.2, the function r ′ is Lipschitz at each point of some set Λ ⊂ R with countablecomplement R \ Λ. The strict convexity of X implies that the phase shift ϕ is a bijective function.The bijectivity of ϕ implies that the set ϕ − (Λ) has countable complement in R and hence is offull measure. Then the set Ω = ϕ − (Λ) ∩ Ω r ′′ ∩ Ω P ∩ Ω T has a full measure and we can applyLemmas 8.2 and 8.7 to see that for any s ∈ Ω the derivative ψ ′ ( s ) can be calculated by the formulagiven in Lemma 8.5. (cid:3) Lemma 8.6. If the Banach space X is AC -smooth, then Z b + Lb ρ ( s ) ds > for every b ∈ R . SOMETRIES BETWEEN SMOOTH 2-DIMENSIONAL BANACH SPACES 19 Proof. Assuming that R b + Lb ρ ( s ) ds = 0 and taking into account that ρ ( s ) ≥ ρ ( s ) = 0 for almost all s ∈ [ b, b + L ]. Taking into account that ρ ( s + L ) = ρ ( s )for every s ∈ ¨Ω r , we conclude that ρ ( s ) is zero almost everywhere. By Lemma 7.1, | τ ( s ) | ≤ ( C + c ) Cc ρ ( s ) = 0 and hence τ ( s ) = 0 almost everywhere. Then also r ′′ = − ρ r − τ r ′ is zero almosteverywhere and by the local absolute continuity of r ′ , the function r ′ is constant and r is containedin a line, which is not true. This contradiction shown that R b + Lb ρ ( s ) ds > (cid:3) Lemma 8.7. If the Banach space X is AC -smooth, then for any b ∈ R , Z b + Lb τ ( s ) ds = 0 . Proof. Let e ∗ , e ∗ : X → R be the biorthogonal linear functionals to the basis e , e , which meansthat e ∗ ( e ) = 1 = e ∗ ( e ) and e ∗ ( e ) = 0 = e ∗ ( e ).Consider the functions x = e ∗ ◦ r and y = e ∗ ◦ r , and observe that r = x · e + y · e and r ′ = x ′ · e + y ′ · e . The linear independence of the vectors r and r ′ implies that the Wronskian W = (cid:12)(cid:12)(cid:12)(cid:12) x yx ′ y ′ (cid:12)(cid:12)(cid:12)(cid:12) = xy ′ − x ′ y of the functions x, y does not take the vaue zero. The AC -smoothness of the Banach space X andLemma 3.2 imply that local absolute continuity of the vector-function r , its coordinate functions x and y , their Wronskian W and its logarithm ln | W | .Consider the derivative of W : W ′ = ( xy ′ − x ′ y ) ′ = x ′′ y + x ′ y ′ − x ′ y ′ − xy ′′ = − ( τ x ′ + ρx ) y + x ( τ y ′ + ρy ) = τ · W. Since W is nowhere equal to zero, this implies (ln | W | ) ′ = W ′ W = τ and by the local absolutecontinuity of ln | W | , Z b + Lb τ ( x ) dx = ln | W ( b + L ) | − ln | W ( b ) | =ln | x ′ ( b + L ) y ( b + L ) − x ( b + L ) y ′ ( b + L ) | − ln | x ′ ( b ) y ( b ) − x ( b ) y ′ ( b ) | =ln | ( − x ′ ( b ))( − y ( b )) − ( − x ( b ))( − y ′ ( b )) | − ln | x ′ ( b ) y ( b ) − x ( b ) y ′ ( b ) | = 0 . (cid:3) Our final lemma yields a formula for calculating the tangential curvature of an absolutelysmooth 2-based Banach space. Lemma 8.8. Assume that the -Banach space X is AC -smooth and the derivative ψ ′ ( s ) of itsquotient curvature ψ is known for almost all points s ∈ R . Then for any b ∈ R the tangentialcurvature τ of X at any point s ∈ ¨Ω r can be calculated by the formula: τ ( s ) = ρ ( s ) · Z sb ψ ′ ( u ) du − R b + Lb R vb ρ ( s ) · ψ ′ ( u ) du dv R b + Lb ρ ( v ) dv . Proof. By Lemma 7.1, the quotient curvature ψ = TP is locally absolutely continuous and hence ψ ( s ) = ψ ( b ) + Z sb ψ ′ ( t ) dt. By Lemma 7.1, τ ( s ) = ρ ( s ) ψ ( s ) = ρ ( s ) ψ ( b ) + Z sb ρ ( s ) ψ ′ ( t ) dt. To find the value of ψ ( b ), we apply Lemma 8.7 and obtain0 = Z b + Lb τ ( s ) ds = ψ ( b ) · Z b + Lb ρ ( s ) ds + Z b + Lb Z sb ρ ( s ) · ψ ′ ( t ) dt ds and finally ψ ( b ) = − R b + Lb R u ρ ( u ) · ψ ′ ( t ) dt du R b + Lb ρ ( u ) du . (cid:3) Proof of Theorem 1.5 Let f : S X → S Y be an isometry between the spheres of two sufficiently smooth 2-dimensionalBanach spaces X, Y . If one of the spaces X or Y is not strictly convex, then by [13, 3.8], theisometry f extends to a linear isometry ¯ f : X → Y and we are done. So, we assume that theBanach spaces X and Y are strictly convex.Fix bases in the spaces X , Y and consider the natural parametrizations r X : R → S X and r Y : R → S Y of the unit spheres of the 2-based Banach spaces X, Y . By Lemma 3.2, the AC -smoothness of the Banach spaces X, Y implies the local absolute continuity of the naturalparametrizations r X and r Y . Let L X and L Y be the half-lengths of the spheres S X and S Y .Consider the segment I X = [0 , L X ] on the real line. By Lemma 3.1, the restriction r X ↾ I X is anisometry of I X onto the arc A = r X ( I X ) endowed with its intrinsic metric. It follows that theintrinsic distance between the endpoints r X (0) and r X ( L X ) of the arc A equals L X .By the result of Tingley [16], f ( − x ) = − f ( x ) for any element x ∈ S X . Consequently, f ( A )is an arc in S Y with endpoints f ( r X (0)) and f ( r X ( L X )) = f ( − r X (0)) = − f ( r X (0)). Then f ( A ) = r Y ( I Y ) for some segment I Y ⊂ R of length L Y . By Lemma 3.1, the restriction r Y ↾ I Y is an isometry of I Y onto the arc f ( A ) = r Y ( I Y ) endowed with its intrinsic metric. Since theisometry f induces the isometry f ↾ A : A → f ( A ) of the arcs A and f ( A ), endowed with theirintrinsic metrics, the map ˜Φ = ( r Y ↾ I Y ) − ◦ f ◦ r X ↾ I X is an isometry of the interval I X onto theinterval I Y . Consequently, the length L X of the interval I X is equal to the length L Y of theinterval I Y . Moreover, for the isometry ˜Φ : I X → I Y there exist real numbers a ∈ {− , } and b such that ˜Φ( x ) = ax + b for all x ∈ I X . Let Φ : R → R be the isometry of the real line, defined byΦ( x ) = ax + b . Let also L := L X = L Y be the common half-length of the spheres S X and S Y .By Lemma 5.4, for every t ∈ R we have r Y ( t − L ) = − r Y ( t − L + L ) = − r Y ( t ) = r Y ( t + L )and hence r Y ( t + aL ) = r Y ( t + L ). By the result of Tingley [16], f ( − x ) = − f ( x ) for any element x ∈ S X . Then by Lemma 5.4(1), for every t ∈ I X = [0 , L ] we have f ( r X ( t + L )) = f ( − r X ( t )) = − f ( r X ( t )) = − r Y (Φ( t )) = − r Y ( at + b ) = r Y ( at + b + aL ) = r Y ( a ( t + L ) + b ) = r Y (Φ( t + L )) . Consequently, f ◦ r X ↾ [0 , L ] = r Y ◦ Φ ↾ [0 , L ] . Now the 2 L -periodicity of the functions r X and r Y imply that f ◦ r X = r Y ◦ Φ . Let ρ X , τ X be the radial and tangential curvatures of the sphere S X , and ρ Y , τ Y be the radialand tangential curvatures of the sphere S Y . Claim 9.1. ρ X ( s ) = ρ Y ◦ Φ( s ) for almost all s ∈ R .Proof. By Lemma 5.4, the functions r X and r Y have the second derivatives r ′′ X and r ′′ Y at eachpoint of some set ¨Ω ⊂ R of full measure. Then the set Ω = ¨Ω ∩ Φ − ( ¨Ω) has full measure. By SOMETRIES BETWEEN SMOOTH 2-DIMENSIONAL BANACH SPACES 21 Lemma 8.1, for every s ∈ Ω we have ρ X ( s ) = lim ε → − k r X ( s + ε ) − r X ( s + L − ε ) k X ε =lim ε → − k f ◦ r X ( s + ε ) − f ◦ r X ( s + L − ε ) k Y ε =lim ε → − k r Y ◦ Φ( s + ε ) − r Y ◦ Φ( s + L − ε ) k Y ε =lim ε → − k r Y (Φ( s ) + aε ) − r Y (Φ( s ) + aL − aε ) k Y ( aε ) =lim ε → − k r Y (Φ( s ) + aε ) − r Y (Φ( s ) + L − aε ) k Y ( aε ) = ρ Y (Φ( s )) . (cid:3) Let ψ X and ψ Y be the quotient curvatures of the Banach spaces X and Y , respectively. ByLemma 7.1, the functions ψ X and ψ Y are locally absolutely continuous and hence are differentiablealmost everywhere. Claim 9.2. ψ ′ X ( s ) = ψ ′ Y ◦ Φ( s ) for almost all s ∈ R .Proof. If the set ρ − X (0) has zero Lebesgue measure, then the set ρ − Y (0) = Φ( ρ − (0)) has zeroLebesgue measure, and the sufficiently smooth Banach spaces X and Y are A ˘ C -smooth. ByLemma 8.3, there are subsets Ω X , Ω Y ⊂ R of full measure such that the restrictions ψ ′ ↾ Ω X and ψ Y ↾ Ω Y can be found by suitable formulas (given in Lemma 8.3). Since Φ is an isometry of thereal line, the set Ω = Ω X ∩ Φ − (Ω Y ) has full measure and for every s ∈ Ω we have the equalities: φ Y (Φ( s )) = φ Y ( as + b ) = − ε → ε − k r Y ( as + b + ε ) − r Y ( as + b − ε ) k Y ρ Y (Φ( s )) ε = − ε → ε − k r Y ( as + b + aε ) − r Y ( as + b − aε ) k Y ρ X ( s ) ε = − ε → ε − k r Y (Φ( s + ε )) − r Y (Φ( s − ε )) k Y ρ X ( s ) ε = − ε → ε − k f ( r X ( s + ε )) − f ( r X ( s − ε )) k Y ρ X ( s ) ε = φ X ( s ) = − ε → ε − k r X ( s + ε ) − r X ( s − ε ) k X ρ X ( s ) ε = φ X ( s )If the set ρ − X (0) has non-zero Lebesgue measure, then the set ρ − Y (0) = Φ( ρ − X (0)) also has non-zero measure, and the sufficiently smooth Banach spaces X, Y are AC L -smooth. By Lemma 8.5,there are subsets Ω X , Ω Y ⊂ R of full measure such that the restrictions ψ ′ ↾ Ω X and ψ Y ↾ Ω Y canbe found by suitable formulas (given in Lemma 8.5). Replacing Ω X and Ω Y by smaller sets offull measure we can assume that for any s ∈ Ω X the real number ρ X ( s ) is well-defined. Since Φis an isometry of the real line, the set Ω = Ω X ∩ Φ − (Ω Y ) has full measure. For every s ∈ Ω with ρ X ( s ) > ρ Y (Φ( s )) = ρ X ( s ) > 0. Repeating the above argument, we can show that ψ X ( s ) = ψ Y (Φ( s )).For every s ∈ Ω with ρ X ( s ) = 0 we have ρ Y (Φ( s )) = ρ X ( s ) = 0 and by Lemma 8.5, ψ ′ X ( s ) = − ψ ′ Y (Φ( s )). (cid:3) Claim 9.3. τ X = a · τ Y ◦ Φ .Proof. By the local absolute continuity of the function ψ X and ψ Y , for every s ∈ R we have theequalities ψ X ( s ) = ψ X (0) + Z s ψ ′ X ( t ) dt = ψ X (0) + Z s φ X ( t ) dt and ψ Y (Φ( s )) = ψ Y ( as + b ) = ψ Y ( b ) + Z as + bb φ Y ( t ) dt = ψ Y ( b ) + a Z s φ Y (Φ( t )) dt = ψ Y ( b ) + a Z s φ X ( t ) dt = ψ Y ( b ) + a ( ψ X ( s ) − ψ X (0)) . To find the values ψ X (0) and ψ Y ( b ), we apply Lemma 8.7:0 = Z L τ X ( s ) ds = Z L ρ X ( s ) ψ X ( s ) ds = Z L ρ X ( s ) (cid:0) ψ X (0) + Z s φ X ( t ) dt (cid:1) ds and hence ψ X (0) = − (cid:16) Z L ρ X ( s ) ds (cid:17) − · Z L Z s ρ X ( s ) · φ X ( t ) dtds. By analogy we can show that ψ Y ( b ) = − (cid:16) Z b + aLb ρ Y ( s ) ds (cid:17) − · Z b + aLb Z sb ρ Y ( s ) · φ Y ( t ) dt ds = − (cid:16) a Z L ρ Y ( at + b ) dt (cid:17) − · Z L (cid:16) Z au + bb ρ Y ( au + b ) · φ Y ( t ) dt (cid:17) a du = − a (cid:16) Z L ρ Y (Φ( t )) dt (cid:17) − · Z L (cid:16) Z u ρ Y (Φ( u )) · φ Y ( av + b ) a dv (cid:17) du = − a (cid:16) Z L ρ X ( t ) dt (cid:17) − · Z L Z u ρ X ( u ) · φ X ( v ) dv du = a · ψ X (0) . Finally, we obtain the desired equality: τ X ( s ) = ρ X ( s ) ψ X ( s ) = ρ X ( s ) · (cid:0) ψ X (0) + Z s φ X ( t ) dt (cid:1) = ρ Y (Φ( s )) · (cid:0) a · ψ Y ( b ) + Z s φ Y (Φ( t )) dt (cid:1) = ρ Y (Φ( s )) · ( a · ψ Y ( b ) + Z s φ Y ( at + b ) dt ) = ρ Y (Φ( s )) · ( a · ψ Y ( b ) + a Z as + bb φ Y ( s ) ds ) = a · ρ Y (Φ( s )) · ( ψ Y ( b ) + Z as + bb φ Y ( s ) ds ) = a · ρ Y (Φ( s )) · ψ Y ( as + b ) = a · ρ Y (Φ( s )) · ψ Y (Φ( s )) = a · τ Y (Φ( s )) . (cid:3) Take a linear isomorphism F : X → Y such that F ( r X (0)) = r Y ( b ) and F ( r ′ X (0)) = a · r ′ Y ( b ).Consider the curves x = F ◦ r X : R → Y and y = r Y ◦ Φ : R → Y in the Banach space Y .Observe that for any t ∈ R y ′ ( t ) = r ′ Y (Φ( t )) · Φ ′ ( t ) = a · r ′ Y (Φ( t ))and y ′′ ( t ) = r ′′ Y (Φ( t )) = − ρ Y (Φ( t )) · r Y (Φ( t )) − τ Y (Φ( t )) ρ ′ Y (Φ( t )) = − ρ X ( t ) · y ( t ) − τ X ( t ) · y ′ ( t )Applying the linear operator F to the equation r ′′ X ( t ) = − ρ X ( t ) · r X ( t ) − τ X ( t ) · r ′ X ( t ) , we obtain the equation(4) x ′′ ( t ) = − ρ X ( t ) · x ( t ) − τ X ( t ) · x ′ ( t ) . Therefore, the AC -smooth functions x and y satisfy the same differential equation (4) and havethe same initial positions: x (0) = F ◦ r X (0) = r Y ( b ) = r Y (Φ(0)) = y (0) and x ′ (0) = F ◦ r ′ X (0) = a · r ′ ( b ) = a · r ′ Y (Φ(0)) = y ′ (0) . SOMETRIES BETWEEN SMOOTH 2-DIMENSIONAL BANACH SPACES 23 Now the Uniqueness Theorem [15, 65.2] for solutions of linear differential equations of second order(see also the proof of Theorem 6.2) guarantees that x = y . The choice of the isometry Φ ensuresthat F ◦ r X ( s ) = x ( s ) = y ( s ) = r Y (Φ( s )) = f ◦ r X ( s )for any s ∈ R . Therefore, the isometry f extends to the linear operator F . Taking into accountthat F ( S X ) = f ( S X ) = S Y implies F ( B X ) = B Y , we conclude that F : X → Y is a linearisometry. 10. Acknowledgements The author would like to express his sincere thanks to Olesia Zavarzina (whose interesting talkat the conference [19] attracted the author’s attention to Tingley’s problem), to Vladimir Kadetsfor many inspiring discussions on this problem, and to the Mathoverflow user Fedor Petrov forsuggesting the idea of the proof of Lemma 4.3(2–4). References [1] L. Cheng, Y. 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