On "Regular Landsberg metrics are always Berwald" by Z. I. Szabo

On “Regular Landsb erg metrics are alw a ys Berw ald” b y Z. I. Szab´ o Vladim ir S. Matveev In his pap er [2], Z. I. Szab´ o claimed (Theorem 3.1) that al l sufficiently smo oth L ands- b er g Finsl e r metrics ar e Berwald ; this claim solv es the long-standing “unicorn” prob- lem. Unfortunately , as I explain b elow , the pro of o f the statement has a gap. F ollo wing [2], let us c onsider a smo ot h n − dimensional manifold M with a prop er Finsler metric F : T M → R . The second differen tial o f 1 2 F 2 | T x M will b e denoted by g = g ( x,y x ) and should b e view ed as a Riemannian metric on the punctured tangen t space T x M − { 0 } . F or a smo o th curv e c ( t ) connecting t w o p o in ts a, b ∈ M , we denote by τ : T a M → T b M , τ ( a, y a |{z} ∈ T a M ) = ( b, φ ( y a ) | {z } ∈ T b M ) the Berw ald parallel tra nspo rt along the curve c . F ollowing [1], Z. I. Szab´ o considers the follow ing Riemannian metric g on M canonically constructed b y F by the f o rm ula g ( x ) ( ξ , η ) := Z y x ∈ T x M F ( x,y x ) ≤ 1 g ( x,y x ) ( ξ , η ) dµ ( x,y x ) (1) where ξ , η ∈ T x M are tw o arbitrary ve ctors, a nd the volume fo r m d µ on T x M is g iv en b y dµ ( x,y x ) := p det( g ( x,y x ) ) dy 1 x ∧ · · · ∧ dy n x . Z. I. Sz ab´ o claims that if the Finsler metric F is L andsb er g, the Berwald p ar a l l e l tr ansp ort pr eserves the Riemannian metric g . According t o the definitions in Section 1 2 of [2 ], this claim means that for ev ery ξ , η , ν ∈ T a M g ( a ) ( ξ , η ) = g ( b ) ( d ν φ ( ξ ) , d ν φ ( η )) . (2) This claim is crucial for the pro of; the remaining part of the pro of is made of relativ ely simple standard arg umen ts, and is correct. The claim itself is explained v ery briefly; basically Z. I. Szab´ o writes that, for Landsb erg metrics, the unite ball { y x ∈ T x M | F ( x, y x ) ≤ 1 } , the v olume form dµ , and the metric g ( x,y x ) are preserv ed b y the parallel transp ort, and, therefore, the metric g giv en b y (1) mus t b e preserv ed as w ell. Indeed, fo r Landsb erg metrics, the unite ball and the v olume form dµ ar e preserv ed b y the parallel transp or t . Unfortunately , it seems that the metric g is preserv ed in a sligh tly differen t wa y one needs to prov e the claim. More precisely , plugging (1) in (2), w e obtain Z y a ∈ T a M F ( a,y a ) ≤ 1 g ( a,y a ) ( ξ , η ) dµ ( a,y a ) = Z y b ∈ T b M F ( b,y b ) ≤ 1 g ( b,y b ) ( d ν φ ( ξ ) , d ν φ ( η )) d µ ( b,y b ) . (3) As it is explained for example in Section 2 of [2], f or ev ery Finsler metric, the parallel transp ort preserv es the unite ball: φ ( { y a ∈ T a M | F ( a, y a ) ≤ 1 } ) = { y b ∈ T b M | F ( b, y b ) ≤ 1 } . (4) The condition that F is Landsb erg implies φ ∗ dµ ( a,y a ) = dµ ( b,φ ( y a )) . Th us, Szab´ o’s claim is trivially true if a t ev ery y a ∈ T a M g ( a,y a ) ( ξ , η ) = g ( b,φ ( y a )) ( d ν φ ( ξ ) , d ν φ ( η )) . (5) But the condition t ha t the metric is Landsb erg means tha t g ( a,y a ) ( ξ , η ) = g ( b,φ ( y a )) ( d y a φ ( ξ ) , d y a φ ( η )) (6) only , i.e., (5) coincides with the definition of the Landsb erg metric at the only p oint y a = ν ∈ T a M . Since no explanation why (3) holds is giv en in the pap er, I tend to supp ose that Z. I. Szab´ o ov ersa w the difference b etw een the formulas (5) and (6); an yw a y , at the presen t 2 p oin t , the pro of of Theorem 3.1 in [2 ] is not complete. Unfortunately , I could not get an y explanation from Z. I. Szab´ o b y email. The unicorn problem remains op en un til somebo dy closes the gap, o r presen ts another pro of, or prov es the existence of a countere xample; at the pr esen t p oint I can do neither of these. A cknow le dgement: I thank Deutsc he F o r sc h ungsgemeinsc haft (Priorit y Program 1154 — Global Differen tial Geometry) for partial financial suppor t. References [1] Z. I. Szab´ o: Berwald metrics c on structe d by Cheval ley’s p olynomia ls , arXiv:math.DG/060 1522(2006) [2] Z. I. Szab´ o : R e gular L andsb er g m etrics ar e always Berwald, Ann. Glo b. Anal. Geom. 2 0 08 Vladimir S. Matv eev, Mathematisc hes Institut, F riedric h-Sc h iller Univ ers it¨ at Jena 07737 J ena, German y , matveev @minet.uni -jena.de 3