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 R n +1 • D –connection, ξ – any transversal vector field • D X Y = ∇ X Y + h ( X, Y ) ξ • D X ξ = − 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 ) = ∇ X Y − � ∇ X Y. (1) • K = 0 if and only if the hypersurface is a nondegenerate quadric. – p. 3/37

• conditions for affine normal imply the apolarity condition trK X = 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 ( X i , X i ) = 0 where X i , 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 ) − [ K X , K Y ] 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 � ∇ X Y ∈ E for arbitrary X, Y ∈ E • E is totally umbilical if there exists a vector field H ∈ E ⊥ such that � � ∇ X Y, Z � = � X, Y �� H, Z � (3) for all X, Y ∈ E and Z ∈ E ⊥ . H is called the mean curvature normal of E • if � � ∇ X H, 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 = E 0 ⊕ E 1 of non-trivial subbundles such that E 1 is spherical and its orthogonal complement is autoparallel then the manifold M is locally isometric to a warped product M 0 × f M 1 where M 0 and M 1 are integral manifolds of E 0 and E 1 . – 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 original 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 x 1 : M 1 → R p +1 and x 2 : M 2 → R q +1 with planar curve γ : I → R 2 : t �→ ( γ 1 ( t ) , γ 2 ( t )) the composition is hypersurface of R p + q +2 constructed as x : M = R × M 1 × M 2 → R p + q +2 : ( t, m 1 , m 2 ) �→ ( γ 1 ( t ) x 1 ( m ) , γ 2 ( t ) x 2 ( m )) – p. 9/37

2. of two improper affine hyperspheres M 1 and M 2 in R p +1 and R q +1 with planar curve γ : I → R 2 : t �→ ( γ 1 ( t ) , γ 2 ( t )) . We assume that both M 1 and M 2 are normalized so M 1 : x p +1 = F 1 ( x 1 , . . . , x p ) and M 2 : y q +1 = F 2 ( y 1 , . . . , y q ) , where F 1 : U 1 ⊂ R p → R and F 2 : U 2 ⊂ R q → R both satisfy the Monge-Ampère equation. We construct x : R × U 1 × U 2 → R p + q +2 : ( t, x 1 , . . . , x p , y 1 , . . . , y q ) �→ ( x 1 , . . . , x p , F 1 ( x 1 , . . . , x p ) + γ 1 ( t ) , y 1 , . . . , y q , F 2 ( y 1 , . . . , y q ) + γ 2 ( t )) . – p. 10/37

3. of an improper affine hypersphere in R p +1 M 1 : x p +1 = F 1 ( x 1 , . . . , x p ) , and a proper affine hypersphere y : M 2 → R q +1 with y as affine normal, with a planar curve γ : I → R 2 : t �→ ( γ 1 ( t ) , γ 2 ( t )) . We compose x : R × U × M 2 → R p + q +2 : ( t, x 1 , . . . , x p , y ) �→ ( x 1 , . . . , x p , F 1 ( x 1 , . . . , x p ) + γ 1 ( t ) , γ 2 ( t ) y ) . – p. 11/37

4. of two improper affine spheres M 1 : x p +1 = F 1 ( x 1 , . . . , x p ) and M 2 : y q +1 = F 2 ( y 1 , . . . , y q ) in R p +1 and R q +1 with curve γ : I → R 2 : t �→ ( γ 1 ( t ) , γ 2 ( t )) . We construct x : R × U 1 × U 2 → R p + q +2 : ( t, x 1 , . . . , x p , y 1 , . . . , y q ) �→ ( x 1 , . . . , x p , γ 2 ( t ) + F 1 ( x 1 , . . . , x p ) + γ 1 ( t ) F 2 ( y 1 , . . . , y q ) , γ 1 ( t ) y 1 , . . . , γ 1 ( t ) y q , γ 1 ( 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 R n +2 such that its tangent bundle is an orthogonal sum, with respect to the metric h , of two distributions, a one-dimensional distribution D 1 spanned by a unit vector field T and an n -dimensional distribution D 2 ( n ≥ 2 ), with local orthonormal frame X 1 , X 2 , . . . , X n such that K ( T, T ) = λ 1 T, K ( T, X ) = λ 2 X, ST = µ 1 T, SX = µ 2 X, ∀ X ∈ D 2 . – p. 14/37

Then either M n +1 is an affine hypersphere such that K T = 0 or is affine congruent to one of the following immersions 1. f ( t, x 1 , . . . , x n ) = ( γ 1 ( t ) , γ 2 ( t ) g 2 ( x 1 , . . . , x n )) , where g 2 : R n → R n +1 is a proper affine hypersphere centered at the origin, for γ 1 , γ 2 such that γ 2 � = 0 , γ ′ 1 γ 2 − γ 1 γ ′ 2 � = 0 , and moreover, sgn ( γ ′ 1 γ 2 ) = sgn ( γ ′ 1 γ 2 − γ 1 γ ′ 2 ) = sgn ( γ ′ 1 γ ′′ 2 − γ ′′ 1 γ ′ 2 ) , – p. 15/37

2. f ( t, x 1 , . . . , x n ) = γ 1 ( t ) C ( x 1 , . . . , x n ) + γ 2 ( t ) e n +1 , where C : R n → R n +2 is an improper affine sphere given by C ( x 1 , . . . , x n ) = ( x 1 , . . . , x n , p ( x 1 , . . . , x n ) , 1) , with the affine normal e n +1 , for γ 1 , γ 2 such that sgn ( γ ′ 1 γ ′′ 2 − γ ′′ 1 γ ′ ) = − sgnγ 1 , 2 γ ′ 1 – p. 16/37

3. f ( t, x 1 , . . . , x n ) = C ( x 1 , . . . , x n ) + γ 2 ( t ) e n +1 + γ 1 ( t ) e n +2 where C : R n → R n +2 is an improper affine sphere given by C ( x 1 , . . . , x n ) = ( x 1 , . . . , x n , p ( x 1 , . . . , x n ) , 1) with the affine normal e n +1 , for γ 1 , γ 2 such that sgn ( γ ′ 1 γ ′′ 2 − γ ′′ 1 γ ′ 2 ) = sgnγ ′ 1 . – p. 17/37

Immersions 1, 2 and 3 satisfy the conditions of the theorem. The affine normal ξ is given by �� � � ( γ ′ 1 γ ′′ 2 − γ ′′ 1 γ ′ 2 ) γ ′ n � � � 1 1. ξ = ϕf + α∂ t f , ϕ = ε n +3 � , � ( γ ′ 1 γ 2 − γ 1 γ ′ 2 ) n +1 γ n 2 γ ′ 1 γ 2 ε = sgn , γ ′ 1 γ 2 − γ 1 γ ′ 2 2 − γ ′′ 2 ) + ϕ ′ ( γ 2 − γ 1 ) γ ′ α ( γ ′′ γ ′ 1 2 = 0 , γ ′ γ ′ 1 1 – p. 18/37

�� � � � γ ′ 1 γ ′′ 2 − γ ′′ 1 γ ′ � � 2 2. ξ = ϕe n +1 + α∂ t f , ϕ = ε n +3 � , � γ ′ 3 1 γ n 1 2 − γ ′ 1 ) + ϕ ′ = 0 , ε = − sgnγ 1 , α ( γ ′′ γ ′′ 2 γ ′ 1 �� � � � γ ′ 1 γ ′′ 2 − γ ′′ 1 γ ′ � � 2 3. ξ = ϕe n +1 + α∂ t f , ϕ = n +3 � , � γ ′ 3 1 2 − γ ′ 1 ) + ϕ ′ = 0 . α ( γ ′′ γ ′′ 2 γ ′ 1 from which all the assumptions follow in a straightforward way. – p. 19/37

Proof: TM = D 1 ⊕ D 2 , D 1 ⊥D 2 , D 1 = Span ( T ) where T is a unit vector field and dim D 2 = n ≥ 2 , with orthonormal frame X 1 , X 2 , . . . , X n such that K ( T, T ) = λ 1 T, K ( T, X ) = λ 2 X, ST = µ 1 T, SX = µ 2 X, ∀ X ∈ D 2 . Apolarity condition for K T implies λ 1 + nλ 2 = 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 ( µ 2 − µ 1 ) h ( T, T ) X with T and arbitrary X ′ ∈ D 2 we obtain – p. 21/37

• ( µ 1 − µ 2 ) h ( X, � ∇ T T ) = X ( µ 1 ) • ( T ( µ 2 ) + λ 2 ( µ 2 − µ 1 )) h ( X, X ′ ) = ( µ 2 − µ 1 ) h ( T, � ∇ X X ′ ) • nX ( λ 2 ) = ( n + 2) λ 2 h ( � ∇ T T, X ) • ( λ 1 − 2 λ 2 ) h ( � ∇ X T, X ′ ) − T ( λ 2 ) h ( X, X ′ ) ∇ T T ) = 1 + h ( K ( X, X ′ ) , � 2( µ 2 − µ 1 ) h ( X, X ′ ) (5) – p. 22/37

• Let � ∇ T T = bU , where U ∈ D 2 is a unit vector field. • L If λ 2 = 0 then M is an affine hypersphere with K T = 0 . • proof: K T = 0 follows from the apolarity condition. From (5) we get bh ( K ( X, X ) , U ) = 1 2 ( µ 2 − µ 1 ) and since bh ( K ( T, T ) , U ) = 0 , we get n � n 2( µ 2 − µ 1 ) = b h ( K ( X i , X i ) , U ) = 0 i =1 where { X 1 = U, X 2 , . . . , X n } is ONB, and µ 1 = µ 2 so the hypersurface is an affine – p. 23/37 hypersphere