Surfaces in R 4 with constant principal angles with respect to a - - PowerPoint PPT Presentation

“Surfaces in R4 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

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Summary: Surfaces in R4 with constant principal angles Published in Geom. Dedicata 2012

We study surfaces in R4 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 R4 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 R2. Using a PDE we prove the existence of surfaces with arbitrary constant principal angles.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

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 ≤ θ1 ≤ θ2 ≤ π

2.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Principal Angles

Definition Let V and W be two-dimensional subspaces of R4. The principal angles between V and W, 0 ≤ θ1 ≤ θ2 ≤ π/2, are given by cos θ1 := v1, w1 := max{v, w|v ∈ V, w ∈ W, |v| = |w| = 1}, cos θ2 := v2, w2 := max{v, w|v ∈ V, w ∈ W, v ⊥ v1, w ⊥ w1, |v| = |w| = 1}.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Surface with principal angles

Definition Let Σ be an immersed surface in R4 and let Π ⊂ R4 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 TpM 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 R4: T 2 = S1 × S1 ⊂ R2 × R2 = R4 with its Riemannian product metric induced from the ambient. T 2 is a helix surface with respect to the plane Π12 = {(x1, x2, x3, x4) ∈ R4|x3 = x4 = 0}.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

The work of Franki Dillen and Daniel Kowalczyk

We classify all the surfaces in M2(c1) × M2(c2) for which the tangent space TpM2 makes constant angles with Tp(M2(c1) × p2) (or equivalently with Tp(p1 × M2(c2)) for every point p = (p1, p2) of M2. Here M2(c1) and M2(c2) are 2-dimensional space forms, not both flat. As a corollary we give a classification of all the totally geodesic surfaces in M2(c1) × M2(c2).

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Principal Angles of Compact surfaces

Lema Let Σ2 ⊂ R4 be a compact immersed surface. Let Π be any two dimensional plane in R4. Then there exist p ∈ Σ, depending on Π, such that TpM and Π have a principal angle equal to zero. Proposition If Σ is an immersed compact helix surface in R4 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 S2( √ 2/2) × S2( √ 2/2).

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Example II

Example A helix in R4 which is full and is not a Riemannian product of two curves. Let Π be a two dimensional subspace of R4 and let G the group

• f all isometries of R4 that fixes pointwise Π. So, G is

isomorphic to the group SO(2). Let γ be a connected regular curve in R4, whose tangent lines makes a constant angle with the plane Π. We define an immersed surface Σ in R4 by taking Σ := G · γ, the orbit of γ under the action of G.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

... 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 R4) 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 R4 that fixes pointwise Π.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Structure Equations

Let Σ ⊂ R4 be a surface with constant principal angles with respect to the plane Π ⊂ R4. Let T1, T2 ∈ Γ(TΣ) be the unit principal vectors and let e1, e2 the corresponding frame of Π. Namely, e1 = cos(θ1)T1 + sin(θ1)ξ1 , e2 = cos(θ2)T2 + sin(θ2)ξ2 where Π = span{e1, e2} and ξ1, ξ2 are normal vector fields. Let X be a vector field of Σ. By taking derivatives in both hands: DXe1 = cos(θ1)DXT1 + sin(θ1)DXξ1 = = cos(θ1)∇XT1 − sin(θ1)Aξ1(X) + cos(θ1)α(X, T1) + sin(θ1)∇⊥

X ξ1

and DXe2 = cos(θ2)DXT2 + sin(θ2)DXξ2 = = cos(θ2)∇XT2 − sin(θ2)Aξ2(X) + cos(θ2)α(X, T2) + sin(θ2)∇⊥

X ξ2

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Structure Equations II

DXe1 = df(X)e2 , DXe2 = −df(X)e1 . By taking the normal and tangent components we get cos(θ2)df(X)T2 = cos(θ1)∇XT1 − sin(θ1)Aξ1(X) , − cos(θ1)df(X)T1 = cos(θ2)∇XT2 − sin(θ2)Aξ2(X) (1) sin(θ2)df(X)ξ2 = cos(θ1)α(X, T1) + sin(θ1)∇⊥

X ξ1 ,

− sin(θ1)df(X)ξ1 = cos(θ2)α(X, T2) + sin(θ2)∇⊥

X ξ2

(2)

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

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(T1, T2) = 0 and α(T1, T1) ⊥ α(T2, T2) . Now the first claim follows from Gauss equation and the second from the Ricci equation.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Principal Direction

Then with respect to the frame T1, T2 we have Aξ1 = m1

• Aξ2 =

m2

• α(T1, T1) = m2ξ2,

α(T2, T2) = m1ξ1

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Caracterizations of complete helix surfaces

Theorem Assume Σ ⊂ R4 with constant principal angles and 0 < θ1 < θ2 < π

2 to be complete. Then Σ is totally geodesic.

Theorem Assume Σ ⊂ R4 with constant principal angles θ1 = 0 to be complete and not totally geodesic. Then T1, T2 are parallel vector fields and Σ is an extrinsic product.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Existence

Existence of surfaces Σ2 ⊂ R4 with constant principal angles θ1, θ2 ∈ (0, π

2). Locally Σ2 can be regarded as a graph of a

function F : Π → Π. F(x, y) = (f(x, y), g(x, y)). That is to say, Σ2 ⊂ R4 is locally given parametrically by (x, y) → (x, y, f(x, y), g(x, y)). The metric tensor , restricted to Σ2 in coordinates (x, y). E := ∂x, ∂x = 1 + f 2

x + g2 x

F := ∂x, ∂y = fxfy + gxgy G := ∂y, ∂y = 1 + f 2

y + g2 y.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

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 Σ2 ⊂ R4. Then Σ2 has constant principal angles with respect to the plane Π = span(e1, e2) if and only if the matrix tensor E F F G

• has constant eigenvalues.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Metric Vs Principal Angles

Proposition Let PF(x, y) = (x, y, f(x, y), g(x, y)) be the parametrization of a surface Σ2 ⊂ R4. Then Σ2 has constant principal angles θ1, θ2 with respect to the plane Π = span(e1, e2) if and only if E + G = sec2(θ1) + sec2(θ2) EG − F2 = sec2(θ1) sec2(θ2) Since EG − F2 = 1 + f 2

x + g2 x + f 2 y + g2 y + (fxgy − fygx)2.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

The PDE system

In order to show the existence of surfaces with principal curvatures we have to solve the following PDE system:        2 + f 2

x + g2 x + f 2 y + g2 y

= sec2(θ1) + sec2(θ2) (fxgy − fygx)2 = sec2(θ1) sec2(θ2) + 1 − sec2(θ1) − sec2(θ2) = tan2(θ1) tan2(θ2) By taking square root we get the equivalent system with the next two equations: f 2

x + g2 x + f 2 y + g2 y

= c1 := sec2(θ1) + sec2(θ2) − 2, fxgy − fygx = c2 := tan(θ1) tan(θ2)

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Finding the PDE

Then from the second equation there exists a function λ such that gx =

−fy+λfx f 2

x +f 2 y ,

gy =

fx+λfy f 2

x +f 2 y .

By using the first equation it follows that λ satisfies: 1 + (f 2

x + f 2 y )2 + λ2

f 2

x + f 2 y

= c1 . Set ∆ := f 2

x + f 2 y . Then λ satisfies

λ2 = c1∆ − 1 − ∆2 .

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

f is a solution of the following second order quasi-linear PDE: (−fy +

• c1∆ − 1 − ∆2fx

∆ )y = (fx +

• c1∆ − 1 − ∆2fy

∆ )x, (3) where ∆ := f 2

x + f 2 y .

Reciprocally, if f is a non lineal solution of the above equation then it is possible to construct g such that F(x, y) = (f(x, y), g(x, y)) is a (non linear) symplectomorphism whose Jacobian matrix has constant length.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Proposition A non totally geodesic surface with constant principal angles do exists if and only if there exists a non linear local symplectomorphism of R2 whose Jacobian matrix has constant length. Proof. Assume that such surface do exists and set ψ(x, y) := (

f √c2 , g √c2 ). Then the above system imply that ψ is a

non linear local symplectomorphism of R2 whose Jacobian matrix has constant length. The converse follows from the above equivalences.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles

Theorem Let Lf = 0 be a second order quasi-linear equation with analytic coefficients. Namely, Lf = A(fx, fy)fxx + B(fx, fy)fxy + C(fx, fy)fyy + E(fx, fy) = 0 , where A, B, C, E are real analytic functions defined in some

• pen subset Ω ⊂ R2.

Then there exists a non linear solution f : D ⊂ R2 → R, i.e. there exists a function f with non constant gradient such that Lf = 0.

Gabriel Ruiz Hern´ andez Surfaces in R4 with constant principal angles