What do we know about intrinsic metric curvature of affine - PDF document

What do we know about intrinsic metric curvature of affine hypersurfaces? Udo Simon Leuven August 29, 2012

Euclidean hypersurface theory: intrinsic curvature has obvious geometric meaning . Affine (relative) hypersurface theories: far from a geometric understanding of intrinsic (metric) curvature 1

Review of relative hypersurfaces non-degenerate scalar product � : R ( n +1) ∗ × R n +1 → R . � , ∇ canonical flat connection on R n +1 hypersurface M conn., or,, diff. mfd, dim. n ≥ 2 x : M → R n +1 immersion normalization: pair ( Y, z ) with � Y, z � = 1 z : M → R n +1 transversal field Y : M → R ( n +1) ∗ conormal field, � Y, dz ( v ) � = 0 . ( x, Y, z ) normalized hypersurface Induced volume forms ω ( v 1 , ..., v n ) := det ( dx ( v 1 ) , ..., dx ( v n ) , z ) ( v 1 , ..., v n ) local frame ω ∗ ( v 1 , ..., v n ) := det ∗ ( dY ( v 1 ) , ..., dY ( v n ) , − Y ) might be trivial 2

Structure equations for ( x, z ) Gauß and Weingarten ∇ v dx ( w ) = dx ( ∇ v w ) + h ( v, w ) z dz ( v ) = dx ( − S ( v )) + τ ( v ) z. Non-degenerate hypersurfaces ( x, Y, z ) non-degenerate ⇔ rank h = n conformal class C ω ∗ is non-degenerate Structure equations for Y ∇ v dY ( w ) = dY ( ∇ ∗ n − 1 Ric ∗ ( v, w )( − Y ) 1 v w )+ 3

Relative normalizations x non-deg. and ( Y, y ) distinguished: dy ( v ) = dx ( − S ( v )) ( x.Y.y ) relative hypersurface Lemma. ( x, Y, y ) rel. hypers. Then • ∇ torsion free and Ricci symmetric; • h semi-Riemannian metric; • shape op. S is h -self adjoint; • ω · ω ∗ = ω 2 (on a frame). 4

Cubic form and Tchebychev form difference tensor C ( v, w ) := ∇ ( h ) v w − ∇ ∗ v w associated cubic form C ♭ ( u, v, w ) := h ( u, C ( v, w )) , totally symmetric Invariants of the pair { h, C } : || C || 2 =: n ( n − 1) J ; Pick invariant trace of C , Tchebychev form T ♭ : n T ♭ ( v ) := trace ( w �→ C ( v, w )) h ( v, T ) := T ♭ ( v ) Tchebychev field Fact: T has a potential function. relative support function x o ∈ R n +1 given fixed point, define: ρ ( x o ) := � Y, x − x o � . 5

Integrability conditions Gauß R ( h ) i C r kj C i rl − C r lj C i = jkl rk � � 1 S lj δ i k − S kj δ i l + h lj S i k − h kj S i + · l 2 Ricci tensor: R ( h ) ij = ij + ( n − 2) C irs C rs − nT r C r S ij + n = 2 Hh ij j 2 Relative Theorema Egregium: n κ ( h ) = J + H − n − 1 h ( T, T ) . 6

Examples of relative normalizations The Euclidean normalization mark “E” for Euclidean invariants µ Euclidean unit normal ( Y ( E ) , y ( E )) = ( µ, µ ) is rel. normalization I , II , III, three fundamental forms h ( E ) = II relative metric 2 C ♭ ( E ) = ∇ ∗ ( III ) II = −∇ ( I ) II cubic form T ♭ ( E ) = − 1 2 n d ln | det S ( E ) | Tchebychev 7

Blaschke normalization ′′ e ′′ as mark y := y ( e ) affine normal , unique within all relative normalizations by T ( e ) = 0 equiv. ω ∗ = ω ( h ), apolarity Centroaffine normalization ′′ c ′′ as mark { p ∈ M | x ( p ) tangential } nowhere dense for x non-degenerate Define: x centroaff. ↔ position vec. always transver. choose rel.normal y ( c ) := ε x , ε = ± 1 Y ( c ) oriented s. t.: 1 = � Y ( c ) , y ( c ) � . Def. x loc. str. convex (i) x hyperbolic type ↔ tangent plane separates origin and hypersurface (ii) x elliptic type ↔ tangent plane does not separate origin and hypersurface 8

Conformal class of relative metrics Fix origin in R n +1 , position vector x transv. q = ρ ♯ Y ♯ = q · Y h ♯ = q · h, ρ ∈ C ∞ and Relate different relative geometries of x : (i) Blaschke - Euclidean: 1 h ( e ) = | det S ( E ) | − n +2 · II (ii) Blaschke - centroaffine: h ( e ) = ρ ( e ) · h ( c ) Calculate Tchebychev forms: (a) Euclidean: 2 n T ( E ) ♭ = − d ln | det S ( E ) | (b) centroaffine: T ( c ) ♭ = n +2 2 n d ln | ρ ( e )( O ) | 9

Gauge invariance invariants independent of particular relative normalization: (i) conformal curvature tensor W Weyl (ii) projective curvature tensor P Weyl for ∇ ∗ : we have P = 0 . � (iii) C ( v, w ) := n n +2 ( T ♭ ( v ) w + T ♭ ( w ) v + h ( v, w ) T ) C ( v, w ) − T ♭ := T ♭ + n +2 (iv) � 2 n d ln ρ ( O ) . 10

Examples: Special classes of hypersurfaces Quadrics ( x, Y, y ) hyperquadric if and only if � C = 0 . Affine (relative) spheres ( x, Y, y ) proper relative sphere if y = λ ( x − x o ) for some x o ∈ R n +1 Fact. ( x, Y, y ) prop. aff. sphere ⇔ ρ ( e )( x o ) = const ⇔ T ( c ) = 0 11

Extremal Blaschke hypersurfaces Euler-Lagrange equ: traceS = 0 nonlinear PDE of fourth order, Monge-Amp` ere type maximal hypersurfaces : Sec. variation area functional negative if: x loc. str. convex, critical point maximal if (i) x has dimension n = 2; or (ii) x is a graph hypers. in dim. n ≥ 2 . Centroaffine extremal hypersurfaces Euler-Lagr. equ: trace ∇ ( h ( c )) T ( c ) = 0 . Example: · · · x α n +1 x α 1 1 x α 2 n +1 = 1 , 2 where α 1 > 0 , · · · , α n +1 > 0; and x 1 , · · · x n +1 pos. canonical coord. R n +1 . 12

Centroaffine Tchebychev hypersurfaces λ ∈ C ∞ ∇ ( h ( c )) T ( c ) = λ · id where Theorem. ( x, Y ( c ) , y ( c )) is centroaffine Tchebychev hypersurface if and only if the equiaffine support function ρ ( e ) satisfies PDE- system Hess ( c ) (ln | ρ ( e ) | ) − 1 n ∆( c )(ln | ρ ( e ) | ) · h ( c ) = 0; PDE independent of choice origin. Corollary. Assume centroaffine Tcheby- chev hypersurface, n ≥ 3 , has complete centroaffine metric. Then it is a proper affine sphere or its metric is conformally flat. 13

Blaschke hypersurfaces Local classification affine spheres, constant sectional curvature n = 2, • l.s. convex surface: quadric (ellipsoid, ell. paraboloid, two-sheeted hyperboloid) • l.s. convex surface: x 1 x 2 x 3 = 1 • indefinite: κ > 0 and H > 0 : ruled surface x = u 1 f ( u 2 ) + f ′ ( u 2 ) • indef: κ = 0 and H = 0 : x 3 = x 1 x 2 + Φ( x 2 ) • indef: κ = 0 and H < 0 : [( x 1 ) 2 + ( x 2 ) 2 ] · x 3 = 1 • indef: κ < 0 and H < 0 : ruled surface x = u 1 f ( u 2 ) + f ′ ( u 2 ) . 14

Local classification affine spheres, constant sectional curvature c : n ≥ 2 and J � = 0 : • l.s. c.: quadric or x 1 x 2 · · · x n +1 = 1 • indef: n = 2 m − 1 and c = 0 ( x 2 1 ± x 2 2 )( x 2 3 ± x 2 4 ) · · · ( x 2 2 m − 1 ± x 2 2 m ) = 1 • indef: n = 2 m and c = 0 ( x 2 1 ± x 2 2 )( x 2 3 ± x 2 4 ) · · · ( x 2 2 m − 1 ± x 2 2 m ) × × x 2 m +1 = 1 15

Inner curvature: Euclidean form II κ ( II ) = H ( E ) + � � 1 ||∇ ( I ) II || 2 II − || d ln | det S ( E ) | || 2 + II 4 n ( n − 1) Global: Let x C 4 -hyperovaloid, κ ( II ) = const. Then x ( M ) Euclidean sphere. Let x ovaloid, κ ( II ) = const · K ( I ) . Then x sphere. Let x ovaloid, κ ( II ) = H ( E ) . Then x sphere 16

Intrinsic curvature: centroaffine Flat centroaffine metric: Canonical centroaffine hypersurfaces x centroaffine hypersurf.; h ( c ) flat, ∇ ( h ( c )) C ( c ) = 0 . Then: · · · x α n +1 (i) x α 1 1 · x α 1 2 n +1 = 1 , where 2 � α 1 � = 0 , 0 < α i for 2 < i, i α i � = 0 . (ii) Let α 2 1 + α 2 2 � = 0 , 0 < α i for 3 ≤ i, 0 � = 2 α 2 + � n +1 and α i . 3 ( − α 1 arctg x 1 x 2 ) ( x 2 3 .. x α n +1 2 ) α 2 · x α 3 1 + x 2 n +1 = 1 e 1 2 x 1 · ( x 2 2 + · · · + x 2 (iii) x n +1 = ν − 1 ) × . × ( α 2 x 2 + ... + α ν − 1 x ν − 1 ) . − x 1 ( α 1 ln x 1 + · · · + α n ln x n ) , where 2 ≤ ν ≤ n + 1 , 0 < α i for ν ≤ i and 0 � = α 1 + α ν + · · · + α n . 17

Differential inequality for κ ( c ). x l. s. convex Tchebychev h-surface, semi-positive Ric . Then: ∆( κ + � T � 2 ) ≥ 4 κ ( κ − ǫ ) + n +2 λ 2 . 4 n Hyperovaloids x centroaffine hyperovaloid. If κ ( c ) = const then 1 ≤ κ. Ovaloids (i) x centroaffine ovaloid. If κ ( c ) = const then 1 = κ. (ii) x centroaffine analytic ovaloid. If κ ( c ) = const then ellipsoid , M¨ unzner (iii) x Tchebychev ovaloid, κ ( c ) = const , then ellipsoid 18

Complete Tchebychev hypersurfaces x l.s. convex, hyperb. Tchebychev, n ≥ 3 h ( c ) complete, Ric ≥ 0 , κ ( c ) = const Then x hyperbolic affine sphere or (i) from canon. h-surface. Proof uses max. principle Omori-Yau: define: F := ( κ + � T � 2 ) ≥ 0; G := ( F + δ ) − 1 2 > 0 . Calculate ( G · ∆ G ): k ( G · ∆ G )( p k ) = − 1 k ( G 4 · ∆ F )( p k ) ≤ 0 ≤ lim 2 lim k G 4 ( p k ) ≤ − 2 κ ( κ + 1) lim − 2 n 2 ( n − 1) k G 4 ( p k ) λ 2 ( p k ) ≤ 0 . lim n +2 19

The nasty term Algebraic curvature tensors Orthogonal decomposition into (unique) ir- reducible subspaces: A = A 1 ⊕ A 2 ⊕ A 3 A 1 : constant curvature type A 2 : scalar flat A 3 : Ricci flat Ric ( A ) ij = C irs C rs − nT r C ijr j τ ( A ) = || C || 2 − n 2 || T || 2 20

Equation of associativity and topological field theory Local hyperbolic graph surface, asymptotic coordinates ( x, y ). C ijk = − 1 2 · ∂ k ∂ j ∂ i f If x has constant curvature metric then A ( w, v, u, z ) = = ( κ + 1) · ( h ( w, u ) h ( v, z ) − h ( v, u ) h ( w, z )) and nasty term reads: ∂ xxx f · ∂ yyy f − ∂ xxy f · ∂ xyy f = const. Similar equation for convex case. 21