Second CoE Talk at the University of Tokyo p. 1/19 How to recognise the geodesics of a metric connection Michael Eastwood Australian National University
Second CoE Talk at the University of Tokyo p. 2/19 References Robert Bryant, Michael Eastwood, and Maciej Dunajski, Metrisability of two-dimensional projective structures, Jour. Diff. Geom. 83 (2009) 465 499. Michael Eastwood and Vladimir Matveev, Metric connections in projective differential geometry, IMA Volumes 144, Springer Verlag 2007, pp. 339 350. Roger Liouville, Sur les invariantes de certaines équations différentiale, Jour. l École Politechnique 59 (1889) 7 76.
Second CoE Talk at the University of Tokyo p. 3/19 Metric geodesics metric Levi-Civita connection geodesics unparameterised geodesics What is lost? What do we obtain? Recall that g ab Γ d ab = 1 2 gcd ( a g bc + b g ac c g ab ) gives the Levi Civita connection a φ b = a φ b Γ c ab φ c.
Second CoE Talk at the University of Tokyo p. 4/19 Projective equivalence Suppose a is a torsion-free connection. Define ˆ a by ˆ a φ b = a φ b Υ a φ b Υ b φ a. ˆ a is torsion-free. ˆ a has the same unparameterised geodesics as a. Conversely, these two properties force. Definition: a and ˆ a are projectively equivalent. Our questions are now operational: σ : {metrics}? {projective structures}.
Example: σ is not injective σ(g ab ) = σ(constant g ab ) but also (1 y 2 )dx 2 + xy dxdy + (1 x 2 )dy 2 (1 x 2 y 2 ) 2 Beltrami on {x 2 + y 2 < 1} has the same unparameterised geodesics as (1 + y 2 )dx 2 xy dxdy + (1 + x 2 )dy 2 (1 + x 2 + y 2 ) 2 Thales or, indeed, as the flat metrics dx 2 + dy 2 or dx 2 dy 2. Second CoE Talk at the University of Tokyo p. 5/19
Second CoE Talk at the University of Tokyo p. 6/19 Example: σ is not surjective Writing a φ b = a φ b Γ ab c φ c, Γ ab 1 = 1 1 + x 2 x 0 0 x Γ ab 2 = 1 1 + x 2 0 x x 0 is a metric connection but Γ ab 1 = x x + y x + y y Γ ab 2 = x 0 0 y is not projectively equivalent to a metric connection. Why Not?
Second CoE Talk at the University of Tokyo p. 7/19 Another example The connection Γ ab 1 = xy + 2e y + 2ye y + 2x 2 ye y 1 + y + x 2 y x + y 2 x + y 2 x 2(1 + y + x 2 y) Γ ab 2 = 1 1 + y + x 2 y e y 1 + x 2 + 4x + 4xy + 4x 3 y + 4y 2 + 4y 3 + 4y 3 x 2 2(1 + y + x 2 y) e y is projectively equivalent to a metric connection!
Second CoE Talk at the University of Tokyo p. 8/19 Naïve dimension count On a surface Jets = Λ 0 + Λ 1 + 2 Λ 1 + 3 Λ 1 + 4 Λ 1 + rank = 1 + 2 + 3 + 4 + 5 + Therefore k 1 2 3 4 5 6 7 rankj k+1 6 10 15 21 28 36 45 rankj k+1 (metrics) 18 30 45 63 84 108 135 ditto up to scale 17 29 44 62 83 107 134 rankj k (proj. struc.) 12 24 40 60 84 112 144 deficit 1 5 10
Second CoE Talk at the University of Tokyo p. 9/19 Obstructions We expect that J k+1 (metrics up to scale) J k (projective structures) is not surjective for k = 5. Therefore, we expect D(Γ) = polynomial in Γ, Γ,, (5) Γ such that Γ metrisable D(Γ) = 0. For k = 6, expect 5 = #{D(Γ), 1 D(Γ), 2 D(Γ)}+ two more invariants and then 10 < 6 + 6 generically this is enough (real-analytic... ).
Second CoE Talk at the University of Tokyo p. 10/19 Special connections Ricci curvature ( a b b a )X b = R ab X b is not necessarily symmetric. However, ˆ a φ b = a φ b Υ a φ b Υ b φ a implies ˆR [ab] = R [ab] (n + 1) [a Υ b]. Bianchi [a R bc] = 0. Thus, restrict to R ab = R (ab). Definition Connections with symmetric Ricci special.
Second CoE Talk at the University of Tokyo p. 11/19 An overdetermined system Theorem (Liouville,..., Sinjukov, Mikeš,... ) A special connection a is projectively equivalent to a metric connection if and only if the system of PDEs trace-free part of ( a σ bc ) = 0 a σ bc 1 n+1 δ a b d σ cd 1 n+1 δ a c d σ bd for σ ab symmetric has a non-degenerate solution. Proof Check that g ab = det(σ)σ ab works, where det(σ) = ǫ a b ǫ c d σ ac σ bd and a ǫ c d = 0 (OK for special connections).
Second CoE Talk at the University of Tokyo p. 12/19 Prolongation Rewrite as a σ bc = δ b a µ c + δ c a µ b. Flat case as warm-up: 0 = ( a b b a )σ bc = n a µ c δ c a b µ b whence But whence a µ b = δ a b ρ. 0 = ( a b b a )µ b = (n 1) a ρ a ρ = 0.
Second CoE Talk at the University of Tokyo p. 13/19 Tractor connection Curved case: a σ bc a µ b = δ a b µ c + δ a c µ b = δ a b ρ P ac σ bc + 1 n W ac b dσ cd a ρ = 2P ab µ b + 4 n Y abcσ bc c c R ab d = W ab d + δ c a P bd δ c b P ad Y abc = [a P b]c. Reorganize as tractors: σ bc a σ bc δ b a µ c δ c a µ b µ b a a µ b δ b a ρ + P ac σ bc 1 n W ac b dσ cd ρ a ρ + 2P ab µ b 4 n Y abcσ bc cf. Cartan connection.
Second CoE Talk at the University of Tokyo p. 14/19 Surfaces c W ab d = 0 and Cotton-York Y abc is projectively invariant. σ bc a σ bc δ b a µ c δ c a µ b Σ α µ b a a µ b δ b a ρ + P ac σ bc ρ a ρ + 2P ab µ b 2Y abc σ bc Compute curvature (fix a volume form ǫ ab ): σ bc 0 ǫ ab a b µ b = 0 ρ 5Y a µ a + Z ab σ ab where Y a = ǫ bc Y bca and Z ab = (a Y b).
Second CoE Talk at the University of Tokyo p. 15/19 Consequences of metrisability If Γ is metrisable, then a Σ α = 0 for some Σ α 0. Now 0 ǫ ab a b Σ α = 0 Ξ α Σ α = 0 where Ξ α Differentiate once more: 5Y a Z ab. 0 = a (Ξ γ Σ γ ) = ( a Ξ γ )Σ γ (dual connection) 5Y a a Σ γ = 5 a Y c + 2Z ac. a Z cd 5P a(c Y d)
Second CoE Talk at the University of Tokyo p. 16/19 First obstruction Differentiate again: 0 = ( (a b) Ξ γ )Σ γ. The 6 6 matrix Ξγ, a Ξ γ, (a b) Ξ γ is singular. The 5 th order expression D(Γ) det Ξγ, a Ξ γ, (a b) Ξ γ is a projectively invariant obstruction to metrisability.
Second CoE Talk at the University of Tokyo p. 17/19 Example Can compute with MAPLE. Γ ab 1 = x x + y x + y y Γ ab 2 = x 0 0 y is not metrisable because is has non-vanishing primary obstruction:
Second CoE Talk at the University of Tokyo p. 18/19 D(Γ) = 32 19683 e 6y2 3x 2 6xy (54648740x 2 307413592xy 71109308y 2 +959364y 12 +784892430x 12 87474336x 3 y+1133803492x 2 y 2 +1003230736xy 3 +880266528x 10 y 6 42320448x 5 y 9 +217229904x 5 y 7 412345728x 13 y 3 244856520x 14 64595700y 9 x+86106087y 8 x 2 +141717600x 16 +7004232x 2 y 12 933120xy 13 1465143957x 10 +52488y 14 2521404720x 7 y 3 +69984x 2 y 14 +1178017344x 11 y 3 +154546056x 7 y 5 240504696x 5 y 5 885577536x 13 y 166793688x 6 y 8 +362722626x 4 y 6 3682383255x 8 y 2 1132982640x 9 y 3 960771456x 9 y 5 +637151832x 10 y 4 111087468x 6 y 6 +26135136x 6 y 10 +2531159442x 10 y 2 3419365428x 9 y 32904576x 5 y 11 +9803808x 4 y 12 +32691870x 4 y 8 +418345989x 6 +1115929062x 8 3565116y 8 108590703y 6 80492982y 7 x 781618710y 5 x 172461312x 9 y 7 +699140160x 15 y+63073512y 10 x 4 306669024x 8 y 8 12083904y 11 x+135746496x 7 y 9 1059525360x 8 y 4 +3132002700x 11 y 1321920x 3 y 13 +918959904x 14 y 2 1020384738x 6 y 4 +130963392x 11 y 5 469505160x 12 y 2 +109834704x 3 y 7 +151175418x 4 +213452946y 4 +455151744x 7 y 7 121150260x 3 y 9 1323467424x 12 y 4 +4388454282x 6 y 2 +791844546x 3 y 5 2367092292x 3 y 3 1892941254x 5 y+2905986870x 7 y +2283906366x 5 y 3 +1959196446x 4 y 4 1297857849x 4 y 2 1956344445x 2 y 4 +285904686y 6 x 2 111011688x 8 y 6 28451520y 11 x 3 +7720677y 10 +58471578y 10 x 2 +73919880)
THANK YOU Second CoE Talk at the University of Tokyo p. 19/19