Affine hypersurfaces with warped product structure Miroslava Anti - - PowerPoint PPT Presentation

Affine hypersurfaces with warped product structure

Miroslava Anti´ c University of Belgrade joint work with Franki Dillen, Kristof Schoels and Luc Vrancken – p. 1/37

• M n+1 – hypersurface in the affine space Rn+1
• D–connection, ξ – any transversal vector field
• DXY = ∇XY + h(X, Y )ξ
• DXξ = −SX + τ(X)ξ
• X, Y – tangential to M, ∇ – induced connection
• h – second fundamental form, S – shape operator
• τ– transversal connection form.

– p. 2/37

• Let h be non-degenerate and ξ the Blashke

normal

• if M is locally strongly convex we may assume

that h is positive definite

• h – the affine metric on the hypersurface M

∇– Levi Civita connection

• K the symmetric difference tensor

K(X, Y ) = ∇XY − ∇XY. (1)

• K = 0 if and only if the hypersurface is a

nondegenerate quadric.

– p. 3/37

• conditions for affine normal imply the apolarity

condition trKX = 0

• the basic structure equation: ∇h and ∇S are

symmetric and h(X, SY ) = h(Y, SX), R(X, Y )Z = h(Y, Z)SX − h(X, Z)SY. implies h(K(X, Y ), Z) = h(K(X, Z), Y ). (2)

– p. 4/37

• M is locally strongly convex and the apolarity

condition give n

i=1 K(Xi, Xi) = 0

where Xi, for i = 1, . . . , n is a local ONB.

• basic equations in terms of the Levi Civita

connection:

R(X, Y )Z = 1

2(h(Y, Z)SX − h(X, Z)SY +

h(SY, Z)X − h(SX, Z)Y ) − [KX, KY ]Z

∇K(X, Y, Z) − ∇K(Y, X, Z) =

1 2(h(Y, Z)SX−h(X, Z)SY + h(SX, Z)Y −

h(SY, Z)X)

∇S(X, Y ) − ∇S(Y, X) = K(SX, Y ) − K(SY, X)

– p. 5/37

∇ and , – Levi-Civita connection and metric of M.

• E ⊂ TM is autoparallel if

∇XY ∈ E for arbitrary X, Y ∈ E

• E is totally umbilical if there exists a vector field

H ∈ E⊥ such that

• ∇XY, Z = X, Y H, Z

(3) for all X, Y ∈ E and Z ∈ E⊥. H is called the mean curvature normal of E

• if

∇XH, Z = 0 then E is spherical

– p. 6/37

T (Hiepko) If the tangent bundle of the Riemannian manifold M splits into an orthogonal sum TM = E0 ⊕ E1 of non-trivial subbundles such that E1 is spherical and its

• rthogonal complement is autoparallel then the

manifold M is locally isometric to a warped product M0 ×f M1 where M0 and M1 are integral manifolds

• f E0 and E1.

– p. 7/37

• affine hyperspheres–normals are either parallel

(improper) or pass through a fixed point (proper)

• Calabi construction (1971)– two hyperbolic affine

hyperspheres are composed into a new hyperbolic affine hypersphere

• the new hypersphere is the product of the two
• riginal ones and a one-dimensional factor whose

image is a special planar curve

• we generalize the above constructions, by taking

an arbitrary curve. We consider the following generalized Calabi constructions:

– p. 8/37

• 1. of two proper affine hyperspheres x1 : M1 → Rp+1

and x2 : M2 → Rq+1 with planar curve γ : I → R2 : t → (γ1(t), γ2(t)) the composition is hypersurface of Rp+q+2 constructed as x : M = R × M1 × M2 → Rp+q+2 : (t, m1, m2) → (γ1(t)x1(m), γ2(t)x2(m))

– p. 9/37

• 2. of two improper affine hyperspheres M1 and M2 in

Rp+1 and Rq+1 with planar curve γ : I → R2 : t → (γ1(t), γ2(t)). We assume that both M1 and M2 are normalized so M1 : xp+1 = F1(x1, . . . , xp) and M2 : yq+1 = F2(y1, . . . , yq), where F1 : U1 ⊂ Rp → R and F2 : U2 ⊂ Rq → R both satisfy the Monge-Ampère equation. We construct x : R × U1 × U2 → Rp+q+2 : (t, x1, . . . , xp, y1, . . . , yq) → (x1, . . . , xp, F1(x1, . . . , xp) + γ1(t), y1, . . . , yq, F2(y1, . . . , yq) + γ2(t)).

– p. 10/37

• 3. of an improper affine hypersphere in Rp+1 M1:

xp+1 = F1(x1, . . . , xp), and a proper affine hypersphere y : M2 → Rq+1 with y as affine normal, with a planar curve γ : I → R2 : t → (γ1(t), γ2(t)). We compose x : R × U × M2 → Rp+q+2 : (t, x1, . . . , xp, y) → (x1, . . . , xp, F1(x1, . . . , xp) + γ1(t), γ2(t)y).

– p. 11/37

• 4. of two improper affine spheres

M1 : xp+1 = F1(x1, . . . , xp) and M2 : yq+1 = F2(y1, . . . , yq) in Rp+1 and Rq+1 with curve γ : I → R2 : t → (γ1(t), γ2(t)). We construct x : R × U1 × U2 → Rp+q+2 : (t, x1, . . . , xp, y1, . . . , yq) → (x1, . . . , xp, γ2(t) + F1(x1, . . . , xp) + γ1(t)F2(y1, . . . , yq), γ1(t)y1, . . . , γ1(t)yq, γ1(t)).

– p. 12/37

• This constructions remain valid when replacing a

factor with a point

• We ask the converse question: when can a given

affine hypersurface be decomposed into a generalized Calabi product of an affine sphere and a point? We show the following theorem:

– p. 13/37

T Let M n+1, n ≥ 2, be a locally strongly convex hypersurface of the affine space Rn+2 such that its tangent bundle is an orthogonal sum, with respect to the metric h, of two distributions, a one-dimensional distribution D1 spanned by a unit vector field T and an n-dimensional distribution D2 (n ≥ 2), with local

• rthonormal frame X1, X2, . . . , Xn such that

K(T, T) = λ1T, K(T, X) = λ2X, ST = µ1T, SX = µ2X, ∀X ∈ D2.

– p. 14/37

Then either M n+1 is an affine hypersphere such that KT = 0 or is affine congruent to one of the following immersions

• 1. f(t, x1, . . . , xn) = (γ1(t), γ2(t)g2(x1, . . . , xn)),

where g2 : Rn → Rn+1 is a proper affine hypersphere centered at the origin, for γ1, γ2 such that γ2 = 0, γ′

1γ2 − γ1γ′ 2 = 0, and moreover,

sgn(γ′

1γ2) = sgn(γ′ 1γ2 − γ1γ′ 2) = sgn(γ′ 1γ′′ 2 − γ′′ 1γ′ 2), – p. 15/37

• 2. f(t, x1, . . . , xn) = γ1(t)C(x1, . . . , xn) + γ2(t)en+1,

where C : Rn → Rn+2 is an improper affine sphere given by C(x1, . . . , xn) = (x1, . . . , xn, p(x1, . . . , xn), 1), with the affine normal en+1, for γ1, γ2 such that sgn(γ′

1γ′′ 2 −γ′′ 1 γ′ 2

γ′

1

) = −sgnγ1,

– p. 16/37

• 3. f(t, x1, . . . , xn) =

C(x1, . . . , xn) + γ2(t)en+1 + γ1(t)en+2 where C : Rn → Rn+2 is an improper affine sphere given by C(x1, . . . , xn) = (x1, . . . , xn, p(x1, . . . , xn), 1) with the affine normal en+1, for γ1, γ2 such that sgn(γ′

1γ′′ 2 − γ′′ 1γ′ 2) = sgnγ′ 1. – p. 17/37

Immersions 1, 2 and 3 satisfy the conditions of the

• theorem. The affine normal ξ is given by
• 1. ξ = ϕf + α∂tf, ϕ = ε

n+3

• (γ′

1γ′′ 2 − γ′′ 1γ′ 2)γ′n 1

(γ′

1γ2 − γ1γ′ 2)n+1γn 2

• ,

ε = sgn γ′

1γ2

γ′

1γ2 − γ1γ′ 2

, α(γ′′

2 − γ′′ 1

γ′

1

γ′

2) + ϕ′(γ2 − γ1)γ′ 2

γ′

1

= 0,

– p. 18/37

• 2. ξ = ϕen+1 + α∂tf, ϕ = ε

n+3

• γ′

1γ′′ 2 − γ′′ 1γ′ 2

γ′3

1 γn 1

• ,

ε = −sgnγ1, α(γ′′

2 − γ′ 2

γ′

1

γ′′

1) + ϕ′ = 0,

• 3. ξ = ϕen+1 + α∂tf, ϕ =

n+3

• γ′

1γ′′ 2 − γ′′ 1γ′ 2

γ′3

1

• ,

α(γ′′

2 − γ′ 2

γ′

1

γ′′

1) + ϕ′ = 0.

from which all the assumptions follow in a straightforward way.

– p. 19/37

Proof: TM = D1 ⊕ D2, D1⊥D2, D1 = Span(T) where T is a unit vector field and dimD2 = n ≥ 2, with

• rthonormal frame X1, X2, . . . , Xn such that

K(T, T) = λ1T, K(T, X) = λ2X, ST = µ1T, SX = µ2X, ∀X ∈ D2. Apolarity condition for KT implies λ1 + nλ2 = 0. (4)

– p. 20/37

By multiplying the identities (∇S)(T, X) = (∇S)(X, T) and ( ∇K)(X, T, T) − ( ∇K)(T, X, T) =

1 2(µ2 − µ1)h(T, T)X

with T and arbitrary X′ ∈ D2 we obtain

– p. 21/37

• (µ1 − µ2)h(X,

∇TT) = X(µ1)

• (T(µ2) + λ2(µ2 − µ1))h(X, X′) =

(µ2 − µ1)h(T, ∇XX′)

• nX(λ2) = (n + 2)λ2h(

∇TT, X)

• (λ1 − 2λ2)h(

∇XT, X′) − T(λ2)h(X, X′) +h(K(X, X′), ∇TT) = 1 2(µ2 − µ1)h(X, X′) (5)

– p. 22/37

• Let

∇TT = bU, where U ∈ D2 is a unit vector field.

• L

If λ2 = 0 then M is an affine hypersphere with KT = 0.

• proof: KT = 0 follows from the apolarity

condition. From (5) we get bh(K(X, X), U) = 1

2(µ2 − µ1)

and since bh(K(T, T), U) = 0, we get n 2(µ2 − µ1) = b

n

• i=1

h(K(Xi, Xi), U) = 0 where {X1 = U, X2, . . . , Xn} is ONB, and µ1 = µ2 so the hypersurface is an affine hypersphere

– p. 23/37

• From now on λ2 = 0.
• From (5) we have −(n + 2)λ2

∇XT = −b n

j=1 h(K(X, Xj), U)Xj + (T(λ2) + 1 2(µ2 −

µ1))X.

• For Xi ∈ D2 orthogonal to U we have

h(( ∇K)(Xi, U, T), Xi) − h(( ∇K)(U, Xi, T), Xi) = 0 −U(λ2) − h(K(U, Xi), ∇XiT) + h(K(Xi, Xi), ∇UT) = 0.

– p. 24/37

• Summing over ONB we obtain

0 = −(n−1)U(λ2)−n

i=2 h(K(U, Xi),

∇XiT)− h(K(T, T), ∇UT) − h(K(U, U), ∇UT)

• We get (n − 1)(n + 2)λ2U(λ2) +

b n

i,j=1 h(K(Xi, U), Xj)2 = 0 – p. 25/37

Denote by K⊥(Xi, U) the projection of the K(Xi, U)

• n the distribution D2 it follows

n

j=1 h(K(Xi, U), Xj)2 = K⊥(Xi, U)2. Then

b((n − 1)(n + 2)2 n λ2

2 + n

• i=1

K⊥(Xi, U)2) = 0. and λ2 = 0 so b = 0. Finally ∇TT = 0 and further X(µ1) = X(µ2) = X(λ2) = 0, ∀X ∈ D2.

– p. 26/37

• We denote by

α = 1 (n + 2)λ2 (T(λ2) + 1 2(µ2 − µ1)). (6)

• L

∇XT = −αX, ∀X ∈ D2.

• Computing h(

R(X, T)T, X) and R(X, Xi)T gives us L T(α) = α2 + 1 2(µ1 + µ2) − λ1λ2 + λ2

2,

(7) X(α) = 0, ∀X.

– p. 27/37

• D1 is autoparallel
• Taking H = αT we get that D2 is spherical
• They satisfy the conditions for the theorem of

Hiepko so M is a warped product M = I ×ρ N, where D1 and D2 are, respectively tangent bundles of I and N

• Let ∂

∂t = T and from now on we denote the

position vector field by f.

– p. 28/37

Case µ2

2 + (α − λ2)2 = 0

• Let β1(t) and β2(t) be functions such that

β′

1 = −β2,

β′

2 = 1 + β1µ1 − β2λ1.

(8)

• denote δ = 1 + β1µ2 + β2α − β2λ2. Since

δ′ = (α − λ2)δ we may chose initial conditions so that that δ vanishes identically

• DX(β1ξ + β2T) = X = DXf, ∀X ∈ D2,

DT(β1ξ + β2T) = T = DTf, so, up to a translation constant, which we may assume vanishes, we can write f = β1ξ + β2T. (9)

– p. 29/37

• g1 = (λ2 − α)ξ + µ2T,

g2 = a(ξ + (α + λ2)T) where function a(t) is such that a′ = −a(α + λ2), a = 0.

• DXg1 = 0, ∀X ∈ D2
• DTg1 = (λ2 − α)′ξ + (λ2 − α)DTξ + µ′

2T +

µ2DTT = (α − (n + 1)λ2)g1, so g1 = φ(t)C

• DTg2 = 0,
• DXg2 = a(−µ2 + λ2

2 − α2)X,

• D

XDXg2 = a(−µ2 + λ2 2 − α2)(

∇

XX⊥ +

K⊥(X, X) + h(X, X)((α + λ2)T + ξ)) belongs to a hyperplane

– p. 30/37

subcase µ2 = λ2

2 − α2

• K⊥ satisfies the apolarity condition so g2 is a

proper affine sphere lying in an affine hyperplane transversal to C

• since µ2 = λ2

2 − α2 we can express ξ and T from

g1 and g2 and finally obtain f(t, x1, . . . , xn) = γ1(t)C + γ2(t)g2(x1, . . . , xn), where (x1, . . . , xn) are the coordinates on N. This yields the first case of the theorem

– p. 31/37

subcase µ2 = λ2

2 − α2

• g2 is a constant vector field
• ξ = 1

β1(f − β2T) and further 1 ag2 = 1 β1f + (α + λ2 − β2 β1)DTf

• by integrating

f(t, x1, . . . , xn) = γ1(t)g(x1, . . . , xn) + γ2(t)g2, and we may assume g2 = (0, . . . , 0, 1, 0)

• D

XDXg = γ1(t)−1(

∇X X⊥ + K⊥(X, X) + h(X, X)(ξ + (α + λ2)T))

– p. 32/37

We can choose initial conditions at a point p0 such that span{Xi(p0)} has as equations xn+1 = xn+2 = 0, then g is contained in the hyperplane xn+2 = 1. We now look at the first n + 1 components of g. Since ξ + (α + λ2)T is collinear to g2 and K⊥ satisfies the apolarity condition it follows that first n + 1 components of g that it is an improper affine sphere with normal vector field g2 from which we obtain case 2 of the theorem.

– p. 33/37

Case µ2

2 + (α − λ2)2 = 0

• Let β1(t) be a function such that β′

1 = −2λ2β1

• DT(β1ξ + 2λ2β1T) = DX(β1ξ + 2λ2β1T) = 0
• β1ξ + 2λ2β1T = C = const and

ξ = 1

β1(C − 2λ2β1T) – p. 34/37

• DTT = K(T, T) + ξ = λ1T + ξ =

(λ1 − 2λ2)T + 1

β1C

• f(t, x1, . . . , xn) =

C2(x1, . . . , xn) + γ1(t)C3(x1, . . . , xn) + γ2(t)C

• γ′′

1 = (λ1 − 2λ2)γ′ 1

γ′′

2 = (λ1 − 2λ2)γ′ 2 + 1 β1 and γ1 = const – p. 35/37

• DXDTf = (λ2 − α)X = 0

DXDTf = γ′

1DXC3, ∀X ∈ D2

• D

XDXf = D XDXC2

= ∇

XX⊥ + K⊥(X,

X) + h(X, X)(2λ2T + ξ)

• K⊥ satisfies the apolarity condition and C2 is an

improper affine sphere in an hyperplane and we

• btain case (3) of the theorem.

– p. 36/37

Thank you

– p. 37/37