“Surfaces in R 4 with constant principal angles with respect to a plane” Gabriel Ruiz Hern´ andez (Joint work with P . Bayard, A. Di Scala y O. Osuna) Instituto de Matem´ aticas, UNAM, Mexico PADGE 2012 Leuven, Belgium August 27 2012 Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Summary: Surfaces in R 4 with constant principal angles Published in Geom. Dedicata 2012 We study surfaces in R 4 whose tangent spaces have constant principal angles with respect to a plane. We classify all surfaces with one principal angle equal to 0. We also classify the complete constant angle surfaces in R 4 with respect to a plane. They turn out to be extrinsic products. The existence of such surfaces turns out to be equivalent to the existence of a special local symplectomorphism of R 2 . Using a PDE we prove the existence of surfaces with arbitrary constant principal angles. Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Linear Algebra of Principal Angles Camille Jordan defined the concept of principal angles between two linear subspaces of the Euclidean space. The principal angles are real numbers between 0 and π 2 which gives a description of the mutual position of two subspaces. If one subspace has dimension one, the principal angle is just the usual angle between a straight line and a subspace. When both subspaces have dimension two, its principal angles consist of two real numbers 0 ≤ θ 1 ≤ θ 2 ≤ π 2 . Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Principal Angles Definition Let V and W be two-dimensional subspaces of R 4 . The principal angles between V and W , 0 ≤ θ 1 ≤ θ 2 ≤ π/ 2, are given by cos θ 1 := � v 1 , w 1 � := max {� v , w �| v ∈ V , w ∈ W , | v | = | w | = 1 } , cos θ 2 := � v 2 , w 2 � := max {� v , w �| v ∈ V , w ∈ W , v ⊥ v 1 , w ⊥ w 1 , | v | = | w | = 1 } . Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Surface with principal angles Definition Let Σ be an immersed surface in R 4 and let Π ⊂ R 4 be a two-dimensional plane. We say that Σ is a helix surface or a constant principal angles surface with respect to Π , if the principal angles between T p M and W do not depend on p ∈ Σ . We will say that Σ has constant principal angles with respect to plane Π . Example (The flat torus is a helix surface) Let us consider the flat torus in R 4 : T 2 = S 1 × S 1 ⊂ R 2 × R 2 = R 4 with its Riemannian product metric induced from the ambient. T 2 is a helix surface with respect to the plane Π 12 = { ( x 1 , x 2 , x 3 , x 4 ) ∈ R 4 | x 3 = x 4 = 0 } . Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
The work of Franki Dillen and Daniel Kowalczyk We classify all the surfaces in M 2 ( c 1 ) × M 2 ( c 2 ) for which the tangent space T p M 2 makes constant angles with T p ( M 2 ( c 1 ) × p 2 ) (or equivalently with T p ( p 1 × M 2 ( c 2 )) for every point p = ( p 1 , p 2 ) of M 2 . Here M 2 ( c 1 ) and M 2 ( c 2 ) are 2-dimensional space forms, not both flat. As a corollary we give a classification of all the totally geodesic surfaces in M 2 ( c 1 ) × M 2 ( c 2 ) . Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Principal Angles of Compact surfaces Lema Let Σ 2 ⊂ R 4 be a compact immersed surface. Let Π be any two dimensional plane in R 4 . Then there exist p ∈ Σ , depending on Π , such that T p M and Π have a principal angle equal to zero. Proposition If Σ is an immersed compact helix surface in R 4 with respect to a plane Π , then it has constant principal angles equal to zero and π/ 2 . That means that its Gauss map image is a product of √ √ two equators in S 2 ( 2 / 2 ) × S 2 ( 2 / 2 ) . Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Example II Example A helix in R 4 which is full and is not a Riemannian product of two curves. Let Π be a two dimensional subspace of R 4 and let G the group of all isometries of R 4 that fixes pointwise Π . So, G is isomorphic to the group SO ( 2 ) . Let γ be a connected regular curve in R 4 , whose tangent lines makes a constant angle with the plane Π . We define an immersed surface Σ in R 4 by taking Σ := G · γ , the orbit of γ under the action of G . Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
... Example II Example Let us observe that Σ is foliated by its geodesics g · γ , for every g ∈ G. The other curves: G · p for every p ∈ γ (these curves are planar circles in R 4 ) are orthogonal to such family of geodesics in Σ . Let us observe that the geodesics on Σ given by g · γ have the same property as the original γ : Their tangent lines make the same constant angle with respect to plane Π , because G consist of isometries in R 4 that fixes pointwise Π . Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Structure Equations Let Σ ⊂ R 4 be a surface with constant principal angles with respect to the plane Π ⊂ R 4 . Let T 1 , T 2 ∈ Γ( T Σ) be the unit principal vectors and let e 1 , e 2 the corresponding frame of Π . Namely, e 1 = cos ( θ 1 ) T 1 + sin ( θ 1 ) ξ 1 , e 2 = cos ( θ 2 ) T 2 + sin ( θ 2 ) ξ 2 where Π = span { e 1 , e 2 } and ξ 1 , ξ 2 are normal vector fields. Let X be a vector field of Σ . By taking derivatives in both hands: D X e 1 = cos ( θ 1 ) D X T 1 + sin ( θ 1 ) D X ξ 1 = = cos ( θ 1 ) ∇ X T 1 − sin ( θ 1 ) A ξ 1 ( X ) cos ( θ 1 ) α ( X , T 1 ) + sin ( θ 1 ) ∇ ⊥ + X ξ 1 and = cos ( θ 2 ) D X T 2 + sin ( θ 2 ) D X ξ 2 = D X e 2 = cos ( θ 2 ) ∇ X T 2 − sin ( θ 2 ) A ξ 2 ( X ) cos ( θ 2 ) α ( X , T 2 ) + sin ( θ 2 ) ∇ ⊥ + X ξ 2 Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Structure Equations II D X e 1 = df ( X ) e 2 , D X e 2 = − df ( X ) e 1 . By taking the normal and tangent components we get cos ( θ 2 ) df ( X ) T 2 = cos ( θ 1 ) ∇ X T 1 − sin ( θ 1 ) A ξ 1 ( X ) , (1) − cos ( θ 1 ) df ( X ) T 1 = cos ( θ 2 ) ∇ X T 2 − sin ( θ 2 ) A ξ 2 ( X ) cos ( θ 1 ) α ( X , T 1 ) + sin ( θ 1 ) ∇ ⊥ sin ( θ 2 ) df ( X ) ξ 2 = X ξ 1 , (2) cos ( θ 2 ) α ( X , T 2 ) + sin ( θ 2 ) ∇ ⊥ − sin ( θ 1 ) df ( X ) ξ 1 = X ξ 2 Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Helix are flat and normal flat Lema The Levi-Civita connection and the normal connection are flat. Proof. Indeed, from above equations it follows α ( T 1 , T 2 ) = 0 and α ( T 1 , T 1 ) ⊥ α ( T 2 , T 2 ) . Now the first claim follows from Gauss equation and the second from the Ricci equation. Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Principal Direction Then with respect to the frame T 1 , T 2 we have � 0 � 0 A ξ 1 = 0 m 1 � m 2 � 0 A ξ 2 = 0 0 α ( T 1 , T 1 ) = m 2 ξ 2 , α ( T 2 , T 2 ) = m 1 ξ 1 Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Caracterizations of complete helix surfaces Theorem Assume Σ ⊂ R 4 with constant principal angles and 0 < θ 1 < θ 2 < π 2 to be complete. Then Σ is totally geodesic. Theorem Assume Σ ⊂ R 4 with constant principal angles θ 1 = 0 to be complete and not totally geodesic. Then T 1 , T 2 are parallel vector fields and Σ is an extrinsic product. Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Existence Existence of surfaces Σ 2 ⊂ R 4 with constant principal angles 2 ) . Locally Σ 2 can be regarded as a graph of a θ 1 , θ 2 ∈ ( 0 , π function F : Π → Π . F ( x , y ) = ( f ( x , y ) , g ( x , y )) . That is to say, Σ 2 ⊂ R 4 is locally given parametrically by ( x , y ) → ( x , y , f ( x , y ) , g ( x , y )) . The metric tensor � , � restricted to Σ 2 in coordinates ( x , y ) . 1 + f 2 x + g 2 := � ∂ x , ∂ x � = E x := � ∂ x , ∂ y � = f x f y + g x g y F 1 + f 2 y + g 2 := � ∂ y , ∂ y � = y . G Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
The Metric of a helix has constant eigenvalues Proposition Let PF ( x , y ) = ( x , y , f ( x , y ) , g ( x , y )) be the parametrization of surface Σ 2 ⊂ R 4 . Then Σ 2 has constant principal angles with respect to the plane Π = span ( e 1 , e 2 ) if and only if the matrix tensor � E � F F G has constant eigenvalues. Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Metric Vs Principal Angles Proposition Let PF ( x , y ) = ( x , y , f ( x , y ) , g ( x , y )) be the parametrization of a surface Σ 2 ⊂ R 4 . Then Σ 2 has constant principal angles θ 1 , θ 2 with respect to the plane Π = span ( e 1 , e 2 ) if and only if sec 2 ( θ 1 ) + sec 2 ( θ 2 ) E + G = EG − F 2 sec 2 ( θ 1 ) sec 2 ( θ 2 ) = Since EG − F 2 = 1 + f 2 x + g 2 x + f 2 y + g 2 y + ( f x g y − f y g x ) 2 . Surfaces in R 4 with constant principal angles Gabriel Ruiz Hern´ andez
Recommend
More recommend
