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

• bvious geometric meaning

. Affine (relative) hypersurface theories: far from a geometric understanding of intrinsic (metric) curvature

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Review of relative hypersurfaces non-degenerate scalar product

• ,

: R(n+1)∗ × Rn+1 → R . ∇ canonical flat connection on Rn+1 hypersurface M conn., or,, diff. mfd, dim. n ≥ 2 x : M → Rn+1 immersion normalization: pair (Y, z) with Y, z = 1 z : M → Rn+1 transversal field Y : M → R(n+1)∗ conormal field, Y, dz(v) = 0. (x, Y, z) normalized hypersurface Induced volume forms ω (v1, ..., vn) := det (dx(v1), ..., dx(vn), z) (v1, ..., vn) local frame ω∗(v1, ..., vn) := det∗(dY (v1), ..., dY (vn), −Y ) might be trivial

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Structure equations for (x, z) Gauß and Weingarten ∇vdx(w) = dx(∇vw) + 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 ∇vdY (w) = dY (∇∗

vw)+ 1 n−1 Ric∗(v, w)(−Y )

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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∗ = ω2 (on a frame).

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Cubic form and Tchebychev form difference tensor C(v, w) := ∇(h)vw − ∇∗

vw

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 xo ∈ Rn+1 given fixed point, define: ρ(xo) := Y, x − xo.

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Integrability conditions Gauß R(h)i

jkl

= Cr

kj Ci rl − Cr lj Ci rk

+

1 2

·

• Sljδi

k − Skjδi l + hlj Si k − hkj Si l

• Ricci tensor:

R(h)ij = = CirsCrs

j

− nTrCr

ij + (n−2) 2

Sij + n

2 Hhij

Relative Theorema Egregium: κ (h) = J + H −

n n−1 h(T, T).

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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 2C♭(E) = ∇∗(III)II = −∇(I)II cubic form T ♭(E) = − 1

2n d ln | det S(E) | Tchebychev

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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

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Conformal class of relative metrics Fix origin in Rn+1, position vector x transv. Y ♯ = q · Y and h♯ = q · h, q = ρ♯

ρ ∈ C∞

Relate different relative geometries of x: (i) Blaschke - Euclidean: h(e) = |det S(E)|−

1 n+2 · II

(ii) Blaschke - centroaffine: h(e) = ρ(e) · h(c) Calculate Tchebychev forms: (a) Euclidean: 2n T(E)♭ = −d ln |det S(E)| (b) centroaffine: T(c)♭ = n+2

2n d ln |ρ(e)(O)|

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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) :=

C(v, w) −

n n+2(T ♭(v)w + T ♭(w)v + h(v, w)T)

(iv) T ♭ := T ♭ + n+2

2n d ln ρ(O).

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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 − xo) for some xo ∈ Rn+1 Fact. (x, Y, y) prop. aff. sphere ⇔ ρ(e)(xo) = const ⇔ T(c) = 0

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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;

• r

(ii) x is a graph hypers. in dim. n ≥ 2. Centroaffine extremal hypersurfaces Euler-Lagr. equ: trace ∇(h(c)) T(c) = 0. Example: xα1

1 xα2 2

· · · xαn+1

n+1 = 1,

where α1 > 0, · · · , αn+1 > 0; and x1, · · · xn+1 pos. canonical coord. Rn+1.

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Centroaffine Tchebychev hypersurfaces ∇(h(c)) T(c) = λ · id where λ ∈ C∞ 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.

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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: x1x2x3 = 1
• indefinite: κ > 0 and H > 0 :

ruled surface x = u1f(u2) + f′(u2)

• indef: κ = 0 and H = 0 :

x3 = x1x2 + Φ(x2)

• indef: κ = 0 and H < 0 :

[(x1)2 + (x2)2] · x3 = 1

• indef: κ < 0 and H < 0 :

ruled surface x = u1f(u2) + f′(u2).

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Local classification affine spheres, constant sectional curvature c: n ≥ 2 and J = 0 :

• l.s. c.: quadric or x1x2 · · · xn+1 = 1
• indef: n = 2m − 1 and c = 0

(x2

1±x2 2)(x2 3±x2 4) · · · (x2 2m−1±x2 2m) = 1

• indef: n = 2m and c = 0

(x2

1 ± x2 2)(x2 3 ± x2 4) · · · (x2 2m−1 ± x2 2m) ×

× x2m+1 = 1

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Inner curvature: Euclidean form II κ(II) = H(E) + +

1 4n(n−1)

• ||∇(I) II||2

II − ||d ln |det S(E)| ||2 II

• Global:

Let x C4-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

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Intrinsic curvature: centroaffine Flat centroaffine metric: Canonical centroaffine hypersurfaces x centroaffine hypersurf.; h(c) flat, ∇(h(c)) C(c) = 0. Then: (i) xα1

1 · xα12 2

· · · xαn+1

n+1 = 1,

where α1 = 0, 0 < αi for 2 < i,

• i αi = 0.

(ii) Let α2

1 + α2 2 = 0, 0 < αi for 3 ≤ i,

and 0 = 2α2 + n+1

3

αi . e

(−α1 arctg x1 x2)(x2 1+x2 2)α2·xα3 3 .. xαn+1 n+1 = 1

(iii) xn+1 =

1 2x1 · (x2 2 + · · · + x2 ν−1)×

. ×(α2x2 + ... + αν−1xν−1) . −x1(α1 ln x1 + · · · + αn ln xn), where 2 ≤ ν ≤ n + 1, 0 < αi for ν ≤ i and 0 = α1 + αν + · · · + αn.

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Differential inequality for κ(c). x l. s. convex Tchebychev h-surface, semi-positive Ric. Then: ∆(κ + T2) ≥ 4κ(κ − ǫ) +

4n n+2λ2.

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

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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 + T2) ≥ 0; G := (F + δ)−1

2 > 0.

Calculate (G · ∆ G): 0 ≤ lim

k (G·∆ G)(pk) = −1 2 lim k (G4·∆ F)(pk) ≤

≤ − 2κ(κ + 1) lim

k G4(pk)

−2n2(n−1)

n+2

lim

k G4(pk)λ2(pk) ≤ 0.

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The nasty term Algebraic curvature tensors Orthogonal decomposition into (unique) ir- reducible subspaces: A = A1 ⊕ A2 ⊕ A3 A1 : constant curvature type A2 : scalar flat A3 : Ricci flat Ric(A)ij = CirsC rs

j

− nT rCijr τ(A) = ||C||2 − n2||T||2

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Equation of associativity and topological field theory Local hyperbolic graph surface, asymptotic coordinates (x, y). Cijk = − 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: ∂xxxf · ∂yyyf − ∂xxyf · ∂xyyf = const. Similar equation for convex case.

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Ejiri’s frame construction Affine spheres, constant curvature metric κ = c; no quadrics Survey four steps. Step 1. unit tangent bundle UM := {(p, v) | p ∈ M, v ∈ TpM, h(v, v) = 1} define F : (p, v) → h(C(v, v)v), UMp compact, F(p, v) max. v = e1. Step 2. operator C(e1, ·) : v → C(e1, v) h-self adjoint, ex. o-normal eigenbasis, say C(e1, ej) = µj · ej. Step 3. extend basis {ei} to local frame, eigenvalues: µ1 = (n − 1) · λ1 > 0, µ2 = · · · = µn = −λ1, and n · λ2 = J. Step 4. C(e1, .) locally is a Codazzi oper- ator; now prove κ = c = 0. As H < 0 we have x1 · x2 · · · xn+1 = 1.

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