New results on the geometry of translation surfaces Marian Ioan - PowerPoint PPT Presentation

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 – 10, 2009 Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 1 / 39

Outline Outline 1 Translation surfaces in E 3 2 On the geometry of the second fundamental form of translation surfaces in E 3 { K II , H } - Generalized Weingarten translation surfaces II -minimality 3 Translation surfaces in the hyperbolic space H 3 4 Translation surfaces in the Heisenberg group Nil 3 5 Translation surfaces in S 3 6 Final remarks Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 2 / 39

Translation surfaces in E 3 Darboux surfaces Cartesian parametrization:    f ( u )   a ( v )  x  = A ( v )  + y g ( u ) b ( v )     z h ( u ) c ( v ) where A ( v ) ∈ O ( n ) 1 Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 3 / 39

Translation surfaces in E 3 Darboux surfaces Cartesian parametrization:    f ( u )   a ( v )  x  = A ( v )  + y g ( u ) b ( v )     z h ( u ) c ( v ) where A ( v ) ∈ O ( n ) A Darboux surface represents a union of ”EQUAL” curves (i.e. the image of one curve 1 , obtained by isometries of the space. 1 generatrix Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 3 / 39

Translation surfaces in E 3 Darboux surfaces A = I 3 : translation surfaces 1 Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 4 / 39

Translation surfaces in E 3 Darboux surfaces A = I 3 : translation surfaces 1 A = matrix of rotation (axe and angle are fixed), a = b = c = 0 : 2 rotation surfaces Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 4 / 39

Translation surfaces in E 3 Darboux surfaces A = I 3 : translation surfaces 1 A = matrix of rotation (axe and angle are fixed), a = b = c = 0 : 2 rotation surfaces A = matrix of rotation (axe ¯ n and angle are fixed), ( a , b , c ) = v ¯ n : 3 helicoidal surfaces Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 4 / 39

Translation surfaces in E 3 Darboux surfaces A = I 3 : translation surfaces 1 A = matrix of rotation (axe and angle are fixed), a = b = c = 0 : 2 rotation surfaces A = matrix of rotation (axe ¯ n and angle are fixed), ( a , b , c ) = v ¯ n : 3 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

Translation surfaces in E 3 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

Translation surfaces in E 3 Tubes r ( s , t ) = γ ( t ) + cos s N ( t ) + sin s B ( t ) Figure: tube r ( s , t ) = γ ( t ) + A ( t ) S 1 Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 5 / 39

Translation surfaces in E 3 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

Translation surfaces in E 3 Translation surfaces If the two curves are situated in orthogonal planes ( x , y , z ) �− → ( x , y , f ( x ) + g ( y )) Examples: planes 1 Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 7 / 39

Translation surfaces in E 3 Translation surfaces If the two curves are situated in orthogonal planes ( x , y , z ) �− → ( x , y , f ( x ) + g ( y )) Examples: planes 1 cylinders 2 Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 7 / 39

Translation surfaces in E 3 Translation surfaces If the two curves are situated in orthogonal planes ( x , y , z ) �− → ( x , y , f ( x ) + g ( y )) Examples: planes 1 cylinders 2 hyperbolic and elliptic paraboloids 3 Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 7 / 39

Translation surfaces in E 3 Translation surfaces If the two curves are situated in orthogonal planes ( x , y , z ) �− → ( x , y , f ( x ) + g ( y )) Examples: planes 1 cylinders 2 hyperbolic and elliptic paraboloids 3 the egg box surface 4 Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 7 / 39

Translation surfaces in E 3 Translation surfaces If the two curves are situated in orthogonal planes ( x , y , z ) �− → ( x , y , f ( x ) + g ( y )) Examples: planes 1 cylinders 2 hyperbolic and elliptic paraboloids 3 the egg box surface 4 Scherk surface 5 Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 7 / 39

Translation surfaces in E 3 Egg box surfaces sin x b + sin y � �� � x , y , a b Figure: egg box surface Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 8 / 39

Translation surfaces in E 3 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

Translation surfaces in E 3 Scherk surface - art ... much more beautiful Figure: Scherk surface Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 10 / 39

Translation surfaces in E 3 Second fundamental form ON THE GEOMETRY OF THE SECOND FUNDAMENTAL FORM OF TRANSLATION SURFACES IN E 3 joint work with A. I. Nistor : arXiv:0812.3166v1 [math.DG] M surface in E 3 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

Translation surfaces in E 3 Second fundamental form ON THE GEOMETRY OF THE SECOND FUNDAMENTAL FORM OF TRANSLATION SURFACES IN E 3 joint work with A. I. Nistor : arXiv:0812.3166v1 [math.DG] M surface in E 3 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

Translation surfaces in E 3 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

Translation surfaces in E 3 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 K II = ⇒ II -flat second mean curvature H II = ⇒ II -minimal Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 12 / 39

Translation surfaces in E 3 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 K II = ⇒ II -flat second mean curvature H II = ⇒ 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 H II vanishes. Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 12 / 39

Translation surfaces in E 3 Old results Koutroufiotis - 1974: a closed ovaloid with K II = cK , c ∈ R or if √ K II = K is a sphere Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 13 / 39

Translation surfaces in E 3 Old results Koutroufiotis - 1974: a closed ovaloid with K II = cK , c ∈ R or if √ K II = K is a sphere Koufogiorgos & Hasanis - 1977: the sphere is the only closed ovaloid satisfying K II = H Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 13 / 39

Translation surfaces in E 3 Old results Koutroufiotis - 1974: a closed ovaloid with K II = cK , c ∈ R or if √ K II = K is a sphere Koufogiorgos & Hasanis - 1977: the sphere is the only closed ovaloid satisfying K II = H Baikoussis & Koufogiorgos - 1997: helicoidal surfaces with ( locally ) K II = H ⇔ constant ratio of the principal curvatures Marian Ioan MUNTEANU (UAIC) On the geometry of translation surfaces Varna, June 2009 13 / 39