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New results on the geometry of translation surfaces Marian Ioan MUNTEANU Al.I.Cuza University of Iasi, Romania webpage: http://www.math.uaic.ro/ munteanu XI th International Conference GEOMETRY, INTEGRABILITY and QUANTIZATION Varna : June 5

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New results on the geometry of translation surfaces

Marian Ioan MUNTEANU

Al.I.Cuza University of Iasi, Romania webpage: http://www.math.uaic.ro/∼munteanu

XIth International Conference GEOMETRY, INTEGRABILITY and QUANTIZATION Varna : June 5 – 10, 2009

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 1 / 39

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Outline

Outline

1 Translation surfaces in E3 2 On the geometry of the second fundamental form of translation surfaces in E3 {KII, H} - Generalized Weingarten translation surfaces II-minimality 3 Translation surfaces in the hyperbolic space H3 4 Translation surfaces in the Heisenberg group Nil3 5 Translation surfaces in S3 6 Final remarks

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 2 / 39

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Translation surfaces in E3

Darboux surfaces

Cartesian parametrization:   x y z   = A(v)   f(u) g(u) h(u)   +   a(v) b(v) c(v)   where A(v) ∈ O(n)

1 Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 3 / 39

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Translation surfaces in E3

Darboux surfaces

Cartesian parametrization:   x y z   = A(v)   f(u) g(u) h(u)   +   a(v) b(v) c(v)   where A(v) ∈ O(n) A Darboux surface represents a union of ”EQUAL” curves (i.e. the image of one curve1, obtained by isometries of the space.

1generatrix Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 3 / 39

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Translation surfaces in E3

Darboux surfaces

1

A = I3 : translation surfaces

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 4 / 39

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Translation surfaces in E3

Darboux surfaces

1

A = I3 : translation surfaces

2

A = matrix of rotation (axe and angle are fixed), a = b = c = 0 : rotation surfaces

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 4 / 39

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Translation surfaces in E3

Darboux surfaces

1

A = I3 : translation surfaces

2

A = matrix of rotation (axe and angle are fixed), a = b = c = 0 : rotation surfaces

3

A = matrix of rotation (axe ¯ n and angle are fixed), (a, b, c) = v¯ n : helicoidal surfaces

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 4 / 39

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Translation surfaces in E3

Darboux surfaces

1

A = I3 : translation surfaces

2

A = matrix of rotation (axe and angle are fixed), a = b = c = 0 : rotation surfaces

3

A = matrix of rotation (axe ¯ n and angle are fixed), (a, b, c) = v¯ n : helicoidal surfaces If the generatrix is

  • a straight line : ruled surfaces
  • a circle : circled surfaces including e.g. tubes

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 4 / 39

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Translation surfaces in E3

Tubes

r(s, t) = γ(t) + cos s N(t) + sin s B(t)

Figure: tube

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 5 / 39

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Translation surfaces in E3

Tubes

r(s, t) = γ(t) + cos s N(t) + sin s B(t)

Figure: tube

r(s, t) = γ(t) + A(t) S1

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 5 / 39

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Translation surfaces in E3

Translation surfaces

Translation surface = ”sum” of two curves

Figure: translation surface

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 6 / 39

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Translation surfaces in E3

Translation surfaces

If the two curves are situated in orthogonal planes (x, y, z) − → (x, y, f(x) + g(y)) Examples:

1

planes

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 7 / 39

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Translation surfaces in E3

Translation surfaces

If the two curves are situated in orthogonal planes (x, y, z) − → (x, y, f(x) + g(y)) Examples:

1

planes

2

cylinders

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 7 / 39

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Translation surfaces in E3

Translation surfaces

If the two curves are situated in orthogonal planes (x, y, z) − → (x, y, f(x) + g(y)) Examples:

1

planes

2

cylinders

3

hyperbolic and elliptic paraboloids

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 7 / 39

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Translation surfaces in E3

Translation surfaces

If the two curves are situated in orthogonal planes (x, y, z) − → (x, y, f(x) + g(y)) Examples:

1

planes

2

cylinders

3

hyperbolic and elliptic paraboloids

4

the egg box surface

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 7 / 39

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Translation surfaces in E3

Translation surfaces

If the two curves are situated in orthogonal planes (x, y, z) − → (x, y, f(x) + g(y)) Examples:

1

planes

2

cylinders

3

hyperbolic and elliptic paraboloids

4

the egg box surface

5

Scherk surface

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 7 / 39

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Translation surfaces in E3

Egg box surfaces

  • x, y, a
  • sin x

b + sin y b

  • Figure: egg box surface

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 8 / 39

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Translation surfaces in E3

Scherk surfaces

  • x, y, a log cos x

a

cos y

a

  • Figure: Scherk surface

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 9 / 39

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Translation surfaces in E3

Scherk surface - art

... much more beautiful

Figure: Scherk surface

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 10 / 39

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Translation surfaces in E3

Second fundamental form

ON THE GEOMETRY OF THE SECOND FUNDAMENTAL FORM OF TRANSLATION SURFACES IN E3 joint work with A. I. Nistor: arXiv:0812.3166v1 [math.DG] M surface in E3 I = the first fundamental form – intrinsic object II = the second fundamental form – extrinsic tool to characterize the twist of M in the ambient

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 11 / 39

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Translation surfaces in E3

Second fundamental form

ON THE GEOMETRY OF THE SECOND FUNDAMENTAL FORM OF TRANSLATION SURFACES IN E3 joint work with A. I. Nistor: arXiv:0812.3166v1 [math.DG] M surface in E3 I = the first fundamental form – intrinsic object II = the second fundamental form – extrinsic tool to characterize the twist of M in the ambient II is a metric if and only if it is non-degenerate curvature properties associated to II:

  • S. Verpoort, The Geometry of the Second Fundamental Form:

Curvature Properties and Variational Aspects,

  • PhD. Thesis, Katholieke Universiteit Leuven, Belgium, 2008

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 11 / 39

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Translation surfaces in E3

Second fundamental form

Lemma (Dillen, Sodsiri - 2005) The second fundamental form II of M is non-degenerate if and only if M is non-developable.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 12 / 39

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Translation surfaces in E3

Second fundamental form

Lemma (Dillen, Sodsiri - 2005) The second fundamental form II of M is non-degenerate if and only if M is non-developable. second Gaussian curvature KII = ⇒ II-flat second mean curvature HII = ⇒ II-minimal

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 12 / 39

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Translation surfaces in E3

Second fundamental form

Lemma (Dillen, Sodsiri - 2005) The second fundamental form II of M is non-degenerate if and only if M is non-developable. second Gaussian curvature KII = ⇒ II-flat second mean curvature HII = ⇒ II-minimal Remark (Verpoort - 2008) Critical points of the area functional of the second fundamental form are those surfaces for which the mean curvature of the second fundamental form HII vanishes.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 12 / 39

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Translation surfaces in E3

Old results

Koutroufiotis - 1974: a closed ovaloid with KII = cK, c ∈ R or if KII = √ K is a sphere

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 13 / 39

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Translation surfaces in E3

Old results

Koutroufiotis - 1974: a closed ovaloid with KII = cK, c ∈ R or if KII = √ K is a sphere Koufogiorgos & Hasanis - 1977: the sphere is the only closed

  • valoid satisfying KII = H

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 13 / 39

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Translation surfaces in E3

Old results

Koutroufiotis - 1974: a closed ovaloid with KII = cK, c ∈ R or if KII = √ K is a sphere Koufogiorgos & Hasanis - 1977: the sphere is the only closed

  • valoid satisfying KII = H

Baikoussis & Koufogiorgos - 1997: helicoidal surfaces with KII = H

(locally)

⇔ constant ratio of the principal curvatures

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 13 / 39

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Translation surfaces in E3

Old results

Koutroufiotis - 1974: a closed ovaloid with KII = cK, c ∈ R or if KII = √ K is a sphere Koufogiorgos & Hasanis - 1977: the sphere is the only closed

  • valoid satisfying KII = H

Baikoussis & Koufogiorgos - 1997: helicoidal surfaces with KII = H

(locally)

⇔ constant ratio of the principal curvatures Blair & Koufogiorgos - 1992: minimal surfaces have vanishing second Gaussian curvature but not conversely

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 13 / 39

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Translation surfaces in E3

Old and recent results

Koutroufiotis - 1974: a closed ovaloid with KII = cK, c ∈ R or if KII = √ K is a sphere Koufogiorgos & Hasanis - 1977: the sphere is the only closed

  • valoid satisfying KII = H

Baikoussis & Koufogiorgos - 1997: helicoidal surfaces with KII = H

(locally)

⇔ constant ratio of the principal curvatures Blair & Koufogiorgos - 1992: minimal surfaces have vanishing second Gaussian curvature but not conversely Kim & Yoon - 2004, Sodsiri - 2005, Yoon - 2006 extends the study for 3-dimensional Lorentz-Minkowski spaces and for different relations between H, K, HII and KII

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 13 / 39

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Translation surfaces in E3

II-flat translation surfaces in E3

1

Theorem (Goemans, Van de Woestyne - 2007) If a translation surface in E3

1 parametrized by ¯

x(s, t) = (s, t, f(s) + g(t)) has KII = 0, then f(s) =

  • F −1(s + d)ds

and g(t) =

  • G−1(t + m)dt

with F and G real functions determined by F(x) =

  • x2

ax4+bx2+cdx and G(x) =

  • x2

−ax4+(2a+b)x2−a−b−cdx,

and a, b, c, d s ¸i m real numbers.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 14 / 39

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Translation surfaces in E3

II-flat PT surfaces in E3

polynomial translation surfaces (in short, PT surfaces) : translation surfaces for which f and g are polynomials Theorem (M., Nistor - 2009) There are no II-flat polynomial translation surfaces in E3. Proof.

KII = 1 (|eg| − f 2)2 ✵ ❇ ❇ ❇ ❅ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ − 1

2evv + fuv − 1 2guu 1 2eu

fu − 1

2ev

fv − 1

2gu

e f

1 2gv

f g ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ − ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞

1 2ev 1 2gu 1 2ev

e f

1 2gu

f g ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ✶ ❈ ❈ ❈ ❆

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 15 / 39

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Translation surfaces in E3

II-flat PT surfaces in E3

(cont.) KII = num 4α′β′∆3/2 where num = −2α(u)2α′(u)2β′(v) − 2α′(u)β(v)2β′(v)2+ 2α(u)2α′(u)β′(v)2 + 2α′(u)2β(v)2β′(v)+ 2α′(u)β′(v)2 + 2α′(u)2β′(v)+ α′(u)β(v)β′′(v) + α(u)α′′(u)β′(v)+ α(u)2α′(u)β(v)β′′(v) + α(u)α′′(u)β(v)2β′(v)+ α′(u)β(v)3β′′(v) + α(u)3α′′(u)β′(v).

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 16 / 39

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Translation surfaces in E3

II-flat translation surfaces

example given by Blair & Koufogiorgos - 1992 : II-flat non-minimal translation surfaces, involving power functions, i.e. α = aup and β = bvq with a, b ∈ R, a, b = 0 and p, q ∈ Q. Proposition (M., Nistor - 2009) The only II-flat translation surfaces with f and g power functions can be parametrized by r(u, v) =

  • u, v, c(u

4 3 − v 4 3 )

  • , c ∈ R∗.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 17 / 39

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Translation surfaces in E3 {KII, H} - Generalized Weingarten translation surfaces

KII = H

{A, B} - generalized Weingarten surfaces : Dillen, Sodsiri - 2005

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 18 / 39

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Translation surfaces in E3 {KII, H} - Generalized Weingarten translation surfaces

KII = H

{A, B} - generalized Weingarten surfaces : Dillen, Sodsiri - 2005 Theorem (M., Nistor - 2009) The only translation surfaces with non-degenerate second fundamental form having the property KII = H are given, up to a rigid motion of R3, by r(u, v) =

  • u, v, 2

c log

  • cos cu

2

cos cv

2

  • , c ∈ R∗.

More, we notice the parametrization of a Scherk type surface, so we have KII = H = 0.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 18 / 39

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Translation surfaces in E3 {KII, H} - Generalized Weingarten translation surfaces

KII = λH, λ = 1, 2

Theorem (M., Nistor - 2009) The only {KII, H}–generalized Weingarten translation surfaces with non-degenerate second fundamental form satisfying KII = λH with λ ∈ R \ {1, 2}, are given, up to a rigid motion of R3, by the parametrization r(u, v) =

  • u, v, 1

p log

  • cos(pv + r)

cos(pu + q)

  • , where p = 0 and r, q ∈ R

which represents a Scherk type surface. Moreover KII = H = 0.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 19 / 39

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Translation surfaces in E3 {KII, H} - Generalized Weingarten translation surfaces

KII = 2H

Theorem (M., Nistor - 2009)

The only translation surfaces with non-degenerate second fundamental form having the property KII = 2H are given, up to a rigid motion of R3, by the following parametrizations

i) Case 1.

r(u, v) = ✒ u, v, − ν 2 log ✒ sinh(pu)

1 p2 cos(qv) 1 q2

✓✓ r(u, v) = ✒ u, v, − ν 2 log ✒ cosh(pu)

1 p2 cos(qv) 1 q2

✓✓ Case 2. r(u, v) = ✵ ❅u, v, ν 2 log cos(pu)

1 p2

cos(qv)

1 q2

✶ ❆

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 20 / 39

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Translation surfaces in E3 {KII, H} - Generalized Weingarten translation surfaces

KII = 2H

i) Case 3.

r(u, v) = ✵ ❅u, v, − ν 2 log sinh(pu)

1 p2

sinh(qv)

1 q2

✶ ❆ r(u, v) = ✵ ❅u, v, − ν 2 log cosh(pu)

1 p2

cosh(qv)

1 q2

✶ ❆ r(u, v) = ✵ ❅u, v, − ν 2 log cosh(pu)

1 p2

sinh(qv)

1 q2

✶ ❆ r(u, v) = ✵ ❅u, v, − ν 2 log sinh(pu)

1 p2

cosh(qv)

1 q2

✶ ❆ .

ii)

r(u, v) = (u, v, a(u − u0)2 − a(v − v0)2), a, u0, v0 ∈ R hyperbolic paraboloid.

iii) combinations of the previous functions in (i) and a second order polynomial (as in (ii), for a certain a)

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 21 / 39

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Translation surfaces in E3 {KII, H} - Generalized Weingarten translation surfaces

Figures

r(u, v) = (u, v, log(sinh u cos v)) r(u, v) = (u, v, log(cosh u cos v))

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 22 / 39

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Translation surfaces in E3 {KII, H} - Generalized Weingarten translation surfaces

Figures

r(u, v) =

  • u, v, log cosh u

cosh v

  • r(u, v) =
  • u, v, log sinh u

cosh v

  • Marian Ioan MUNTEANU (UAIC)

On the geometry of translation surfaces Varna, June 2009 23 / 39

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Translation surfaces in E3 II-minimality

II-minimal surfaces

Haesen, Verpoort, Verstraelen - 2008 HII = −H − 1 4∆II log(K) where ∆II is the Laplacian for functions computed with respect to the second fundamental form as metric. HII can be equivalently expressed as HII = −H − 1 2 √ det II

  • i,j

∂ ∂ui √ det II hij ∂ ∂uj (log √ K)

  • .

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 24 / 39

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Translation surfaces in E3 II-minimality

II-minimal translation surfaces

(u, v) → (u, v, f(u) + g(v)); α = f ′, β = g′ HII = 0 is equivalent to (1 + α2)β′ + (1 + β2)α′ − 4 (1 + α2 + β2)2 + α′′′α′ − 2α′′2 2α′4 + β

′′′β′ − 2β′′2

2β′4 = 0

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 25 / 39

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Translation surfaces in E3 II-minimality

II-minimal translation surfaces

(u, v) → (u, v, f(u) + g(v)); α = f ′, β = g′ HII = 0 is equivalent to (1 + α2)β′ + (1 + β2)α′ − 4 (1 + α2 + β2)2 + α′′′α′ − 2α′′2 2α′4 + β

′′′β′ − 2β′′2

2β′4 = 0 After STRAIGHTFORWARD COMPUTATIONS it follows α′ = 0, β′ = 0 which cannot occur since II is no longer invertible

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 25 / 39

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Translation surfaces in E3 II-minimality

II-minimal translation surfaces

(u, v) → (u, v, f(u) + g(v)); α = f ′, β = g′ HII = 0 is equivalent to (1 + α2)β′ + (1 + β2)α′ − 4 (1 + α2 + β2)2 + α′′′α′ − 2α′′2 2α′4 + β

′′′β′ − 2β′′2

2β′4 = 0 After STRAIGHTFORWARD COMPUTATIONS it follows α′ = 0, β′ = 0 which cannot occur since II is no longer invertible Theorem (M., Nistor - 2009) There are NO II-minimal translation surfaces in Euclidean 3-space.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 25 / 39

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Translation surfaces in the hyperbolic space H3

General things

  • R. L´
  • pez : arXiv:0902.4085v1 [math.DG]

H3 hyperbolic space : upper half-space R3

+

ds2 = 1

z2

  • dx2 + dy2 + dz2

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 26 / 39

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SLIDE 46

Translation surfaces in the hyperbolic space H3

General things

  • R. L´
  • pez : arXiv:0902.4085v1 [math.DG]

H3 hyperbolic space : upper half-space R3

+

ds2 = 1

z2

  • dx2 + dy2 + dz2

the absence of an affine structure does not permit to give an intrinsic concept of translation surface as in E3 = ⇒ sum of planar curves

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 26 / 39

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Translation surfaces in the hyperbolic space H3

General things

  • R. L´
  • pez : arXiv:0902.4085v1 [math.DG]

H3 hyperbolic space : upper half-space R3

+

ds2 = 1

z2

  • dx2 + dy2 + dz2

the absence of an affine structure does not permit to give an intrinsic concept of translation surface as in E3 = ⇒ sum of planar curves x, y are interchangeable, but not with z type 1 : r(x, y) = {x, y, f(x) + g(y)} type 2 : r(x, z) = {x, f(x) + g(z), z}

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 26 / 39

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SLIDE 48

Translation surfaces in the hyperbolic space H3

General things

  • R. L´
  • pez : arXiv:0902.4085v1 [math.DG]

H3 hyperbolic space : upper half-space R3

+

ds2 = 1

z2

  • dx2 + dy2 + dz2

the absence of an affine structure does not permit to give an intrinsic concept of translation surface as in E3 = ⇒ sum of planar curves x, y are interchangeable, but not with z type 1 : r(x, y) = {x, y, f(x) + g(y)} type 2 : r(x, z) = {x, f(x) + g(z), z} Notice that there are NO isometries of H3 that carry surfaces of type 1 into surfaces of type 2 or vice-versa.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 26 / 39

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SLIDE 49

Translation surfaces in the hyperbolic space H3

Minimal translation surface

Recall: in E3 = ⇒ planes and Scherk surface Known fact: Examples of minimal surfaces in H3: totally geodesic planes, minimal graphs (corresponding to Dirichlet problem)

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 27 / 39

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SLIDE 50

Translation surfaces in the hyperbolic space H3

Minimal translation surface

Recall: in E3 = ⇒ planes and Scherk surface Known fact: Examples of minimal surfaces in H3: totally geodesic planes, minimal graphs (corresponding to Dirichlet problem) Theorem (L´

  • pez - 2009)

There are NO minimal translation surfaces in H3 of type 1. The only minimal translation surfaces in H3 of type 2 are totally geodesic planes.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 27 / 39

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Translation surfaces in the Heisenberg group Nil3

Nil3

Heisenberg group Nil3 ∼ R3 (x1, y1, z1) · (x2, y2, z2) :=

  • x1 + x2, y1 + y2, z1 + z2 + 1

2 (x1y2 − x2y1)

  • g = dx2 + dy2 +
  • dz + 1

2 (ydx − xdy) 2

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 28 / 39

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SLIDE 52

Translation surfaces in the Heisenberg group Nil3

Nil3

Heisenberg group Nil3 ∼ R3 (x1, y1, z1) · (x2, y2, z2) :=

  • x1 + x2, y1 + y2, z1 + z2 + 1

2 (x1y2 − x2y1)

  • g = dx2 + dy2 +
  • dz + 1

2 (ydx − xdy) 2 rich properties: homogeneous space, the group of isometries has dimension 4, contact Riemannian structure

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 28 / 39

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Translation surfaces in the Heisenberg group Nil3

Nil3

Heisenberg group Nil3 ∼ R3 (x1, y1, z1) · (x2, y2, z2) :=

  • x1 + x2, y1 + y2, z1 + z2 + 1

2 (x1y2 − x2y1)

  • g = dx2 + dy2 +
  • dz + 1

2 (ydx − xdy) 2 rich properties: homogeneous space, the group of isometries has dimension 4, contact Riemannian structure Lie algebra of Iso(Nil3) is generated by Killing v. f. E1 = ∂ ∂x + y 2 ∂ ∂z E2 = ∂ ∂y − x 2 ∂ ∂z E3 = ∂ ∂z E4 = −y ∂ ∂x + x ∂ ∂y

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 28 / 39

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SLIDE 54

Translation surfaces in the Heisenberg group Nil3

Nil3

E4 generates the group of rotations around z-axis ∼ SO(2)

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 29 / 39

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SLIDE 55

Translation surfaces in the Heisenberg group Nil3

Nil3

E4 generates the group of rotations around z-axis ∼ SO(2) G1 = {(t, 0, 0)|t ∈ R}, G2 = {(0, t, 0)|t ∈ R}, G3 = {(0, 0, t)|t ∈ R}

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 29 / 39

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SLIDE 56

Translation surfaces in the Heisenberg group Nil3

Nil3

E4 generates the group of rotations around z-axis ∼ SO(2) G1 = {(t, 0, 0)|t ∈ R}, G2 = {(0, t, 0)|t ∈ R}, G3 = {(0, 0, t)|t ∈ R} Definition (Figueroa, Mercuri, Pedrosa - 1999) A surface in Nil3 is translation invariant if it is invariant under the action

  • f 1-parameter subgroup generated by a Killing vector field of the form

a1E1 + a2E2 + a3E3 , a2

1 + a2 2 + a2 3 = 0.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 29 / 39

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SLIDE 57

Translation surfaces in the Heisenberg group Nil3

Nil3

E4 generates the group of rotations around z-axis ∼ SO(2) G1 = {(t, 0, 0)|t ∈ R}, G2 = {(0, t, 0)|t ∈ R}, G3 = {(0, 0, t)|t ∈ R} Definition (Figueroa, Mercuri, Pedrosa - 1999) A surface in Nil3 is translation invariant if it is invariant under the action

  • f 1-parameter subgroup generated by a Killing vector field of the form

a1E1 + a2E2 + a3E3 , a2

1 + a2 2 + a2 3 = 0.

Proposition (Figueroa, Mercuri, Pedrosa - 1999) Let M in Nil3 be invariant under the 1-parameter group generated by a1E1 + a2E2 + a3E3 , a2

1 + a2 2 = 0.

Then is it equivalent to a surface invariant under G1.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 29 / 39

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SLIDE 58

Translation surfaces in the Heisenberg group Nil3

Flat translation invariant surfaces

translation invariant surfaces : restrict to G1 and G3

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 30 / 39

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SLIDE 59

Translation surfaces in the Heisenberg group Nil3

Flat translation invariant surfaces

translation invariant surfaces : restrict to G1 and G3 Proposition (Inoguchi - 2005) Let M be a surface invariant under G3 = {(0, 0, t) : t ∈ R}. Then M is locally expressed as (0, 0, v) · (x(u), y(u), 0) , u ∈ I, v ∈ R. I - open interval, u - arclength parameter

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 30 / 39

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SLIDE 60

Translation surfaces in the Heisenberg group Nil3

Flat translation invariant surfaces

translation invariant surfaces : restrict to G1 and G3 Proposition (Inoguchi - 2005) Let M be a surface invariant under G3 = {(0, 0, t) : t ∈ R}. Then M is locally expressed as (0, 0, v) · (x(u), y(u), 0) , u ∈ I, v ∈ R. I - open interval, u - arclength parameter Remark 1. (x, y, 0) and (0, 0, v) commute. Remark 2. M is flat

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 30 / 39

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SLIDE 61

Translation surfaces in the Heisenberg group Nil3

Flat translation invariant surfaces

Proposition (Inoguchi - 2005) Let M be a surface invariant under G1 = {(t, 0, 0) , t ∈ R}. Then M is flat if and only if il is locally equivalent to the graph of f(x, y) = xy 2 + 1 2A

  • y
  • y2 − A2 − A2 log |y +
  • y2 − A2|
  • ,

A ∈ R∗. Proof. idea: the translation invariant surface (G1) is locally parametrized as the graph (x, 0, 0) · (0, y, v(y)) =

  • x, y, v(y) + xy

2

  • .

compute K + solve ODE

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 31 / 39

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SLIDE 62

Translation surfaces in the Heisenberg group Nil3

Minimal G1 - invariant surfaces

Proposition (Inoguchi - 2005) Let M be a surfaces invariant under G1 = {(t, 0, 0) , t ∈ R}. Then M is minimal if and only if il is locally equivalent to the graph of f(x, y) = xy 2 + a

  • y
  • 1 + y2 + log(y +
  • 1 + y2)
  • ,

a ∈ R∗.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 32 / 39

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SLIDE 63

Translation surfaces in the Heisenberg group Nil3

Minimal G1 - invariant surfaces

Proposition (Inoguchi - 2005) Let M be a surfaces invariant under G1 = {(t, 0, 0) , t ∈ R}. Then M is minimal if and only if il is locally equivalent to the graph of f(x, y) = xy 2 + a

  • y
  • 1 + y2 + log(y +
  • 1 + y2)
  • ,

a ∈ R∗. Extensions: using translation to the right for curves in the xz-plane and yz-plane : no flatness results

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 32 / 39

slide-64
SLIDE 64

Translation surfaces in the Heisenberg group Nil3

Minimal G1 - invariant surfaces

Proposition (Inoguchi - 2005) Let M be a surfaces invariant under G1 = {(t, 0, 0) , t ∈ R}. Then M is minimal if and only if il is locally equivalent to the graph of f(x, y) = xy 2 + a

  • y
  • 1 + y2 + log(y +
  • 1 + y2)
  • ,

a ∈ R∗. Extensions: using translation to the right for curves in the xz-plane and yz-plane : no flatness results Why nothing about G4?

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 32 / 39

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SLIDE 65

Translation surfaces in the Heisenberg group Nil3

Minimal G1 - invariant surfaces

Proposition (Inoguchi - 2005) Let M be a surfaces invariant under G1 = {(t, 0, 0) , t ∈ R}. Then M is minimal if and only if il is locally equivalent to the graph of f(x, y) = xy 2 + a

  • y
  • 1 + y2 + log(y +
  • 1 + y2)
  • ,

a ∈ R∗. Extensions: using translation to the right for curves in the xz-plane and yz-plane : no flatness results Why nothing about G4? G4 invariant surfaces are nothing but rotational surfaces around z-axis (G4 = SO(2)) Classification results: Caddeo, Piu, Ratto - 1996

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 32 / 39

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SLIDE 66

Translation surfaces in S3

”Sum” of two curves

work in progress with Rafael L´

  • pez

S3 hypersurface in R4 ≡ H (noncommutative field of quaternions) S3 group of unit quaternions α(s), β(t) curves on S3 (parametrized by arclength)

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 33 / 39

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SLIDE 67

Translation surfaces in S3

”Sum” of two curves

work in progress with Rafael L´

  • pez

S3 hypersurface in R4 ≡ H (noncommutative field of quaternions) S3 group of unit quaternions α(s), β(t) curves on S3 (parametrized by arclength) translation surface: r(s, t) = α(s) · β(t)

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 33 / 39

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SLIDE 68

Translation surfaces in S3

”Sum” of two curves

work in progress with Rafael L´

  • pez

S3 hypersurface in R4 ≡ H (noncommutative field of quaternions) S3 group of unit quaternions α(s), β(t) curves on S3 (parametrized by arclength) translation surface: r(s, t) = α(s) · β(t) Example (well known) r(s, t) = (cos s cos t, sin s cos t, cos s sin t, sin s sin t).

  • α = (cos s, sin s, 0, 0), β(t) = (cos t, 0, sin t, 0): translation surface
  • minimal and II-minimal

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 33 / 39

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SLIDE 69

Translation surfaces in S3

Generalities

From now on FIX α(s) = (cos s, sin s, 0, 0).

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 34 / 39

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SLIDE 70

Translation surfaces in S3

Generalities

From now on FIX α(s) = (cos s, sin s, 0, 0). β(t) ∈ S3: ∃ q = q(t) ∈ S2 ⊂ ImH s.t. β′(t) = β(t)q(t)

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 34 / 39

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SLIDE 71

Translation surfaces in S3

Generalities

From now on FIX α(s) = (cos s, sin s, 0, 0). β(t) ∈ S3: ∃ q = q(t) ∈ S2 ⊂ ImH s.t. β′(t) = β(t)q(t) g = ds2 + 2Fdsdt + dt2, F = ir, rq N = jζr , ζ ∈ S1 ⊂ C ad(r)(q), jζ = 0

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 34 / 39

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SLIDE 72

Translation surfaces in S3

Generalities

From now on FIX α(s) = (cos s, sin s, 0, 0). β(t) ∈ S3: ∃ q = q(t) ∈ S2 ⊂ ImH s.t. β′(t) = β(t)q(t) g = ds2 + 2Fdsdt + dt2, F = ir, rq N = jζr , ζ ∈ S1 ⊂ C ad(r)(q), jζ = 0 there exists x ∈ (0, 1) depending on s and t such that N = ± 1 √ 1 − x2 (xr + irq) ad(r)(q) = xi ±

  • 1 − x2 ijζ.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 34 / 39

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SLIDE 73

Translation surfaces in S3

Generalities

From now on FIX α(s) = (cos s, sin s, 0, 0). β(t) ∈ S3: ∃ q = q(t) ∈ S2 ⊂ ImH s.t. β′(t) = β(t)q(t) g = ds2 + 2Fdsdt + dt2, F = ir, rq N = jζr , ζ ∈ S1 ⊂ C ad(r)(q), jζ = 0 there exists x ∈ (0, 1) depending on s and t such that N = ± 1 √ 1 − x2 (xr + irq) ad(r)(q) = xi ±

  • 1 − x2 ijζ.

The function x does not depend on s!!

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 34 / 39

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SLIDE 74

Translation surfaces in S3

First results

Proposition (L´

  • pez, M. - 2009)

The surface S is flat.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 35 / 39

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SLIDE 75

Translation surfaces in S3

First results

Proposition (L´

  • pez, M. - 2009)

The surface S is flat. Example (the easiest: q′ = 0) β(t) = (cos t, sin t sin θ0, sin t cos θ0 cos ψ0, sin t cos θ0 sin θ0). Proof. ∂ ∂t ad(r)(q) = ad(r)(q′) β′(t) = ξ0β(t) ξ0 = sin θ0 i + jw0, w0 ∈ C, |w0| = cos θ0, θ0 ∈

  • −π

2 , π 2

  • .

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 35 / 39

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SLIDE 76

Translation surfaces in S3

First results

Proposition (L´

  • pez, M. - 2009)

The surface S is flat. Example (the easiest: q′ = 0) β(t) = (cos t, sin t sin θ0, sin t cos θ0 cos ψ0, sin t cos θ0 sin θ0). Proof. ∂ ∂t ad(r)(q) = ad(r)(q′) β′(t) = ξ0β(t) ξ0 = sin θ0 i + jw0, w0 ∈ C, |w0| = cos θ0, θ0 ∈

  • −π

2 , π 2

  • .
  • Remark. All these surfaces are minimal.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 35 / 39

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SLIDE 77

Translation surfaces in S3

Other results

Recall N = jζr , ζ ∈ S1 ⊂ C ζ = cos ϕ + sin ϕ i , ϕ = ϕ(s, t) Weingarten operator : A =     − x √ 1 − x2 1 + xϕt √ 1 − x2 1 √ 1 − x2 − x + ϕt √ 1 − x2    

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 36 / 39

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SLIDE 78

Translation surfaces in S3

Other results

Recall N = jζr , ζ ∈ S1 ⊂ C ζ = cos ϕ + sin ϕ i , ϕ = ϕ(s, t) Weingarten operator : A =     − x √ 1 − x2 1 + xϕt √ 1 − x2 1 √ 1 − x2 − x + ϕt √ 1 − x2     Proposition (L´

  • pez, M. - 2009)

The surface S cannot be totally geodesic in S3.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 36 / 39

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SLIDE 79

Translation surfaces in S3

Minimality

Proposition (L´

  • pez, M. - 2009)

The surface S is minimal if and only if ϕ(s, t) = −2

  • s +
  • x(t)dt
  • .

Moreover

ad(r)(q) = x i − ♣ 1 − x2 ✒ − sin ✏ 2 ❩ x(t)dt + 2s ✑ j + cos ✏ 2 ❩ x(t)dt + 2s ✑ k ✓

where x = x(t) is a smooth function.

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 37 / 39

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SLIDE 80

Translation surfaces in S3

Minimality

Proposition (L´

  • pez, M. - 2009)

The surface S is minimal if and only if ϕ(s, t) = −2

  • s +
  • x(t)dt
  • .

Moreover

ad(r)(q) = x i − ♣ 1 − x2 ✒ − sin ✏ 2 ❩ x(t)dt + 2s ✑ j + cos ✏ 2 ❩ x(t)dt + 2s ✑ k ✓

where x = x(t) is a smooth function. Difficulties: In order to give an explicit expression for β we have to solve the following QODE β′(t) = µ(t)β(t) , µ(t) is known

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 37 / 39

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SLIDE 81

Problem

Find a 3-dimensional space and an embedding such that the following

  • bject becomes II-minimal or II-flat

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 38 / 39

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SLIDE 82

Ceramic joke

Find a 3-dimensional space and an embedding such that the following

  • bject becomes II-minimal or II-flat

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 38 / 39

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SLIDE 83

THANK YOU FOR ATTENTION !

Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 39 / 39