As proje c oes de superf Jorge Luiz Deolindo Silva Universidade - PowerPoint PPT Presentation

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio ıcies em R 4 As proje¸ c˜ oes de superf´ Jorge Luiz Deolindo Silva Universidade Federal de Santa Catarina - Blumenau 28/04/2017 Col´ oquio de Matem´ atica - UFSC

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Contents Mochida, Romero-Fuster R. Thom H. Whitney and Ruas and I. Porteous James Montaldi Bruce and T ari We are here 1955 80's 1986 1995 2002 2015

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Contents Mochida, Romero-Fuster R. Thom H. Whitney and Ruas and I. Porteous James Montaldi Bruce and T ari We are here 1955 80's 1986 1995 2002 2015 Singularity Theory

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Contents Mochida, Romero-Fuster R. Thom H. Whitney and Ruas and I. Porteous James Montaldi Bruce and T ari We are here 1955 1986 1995 2002 2015 80's

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Contents Mochida, Romero-Fuster R. Thom H. Whitney and Ruas and I. Porteous James Montaldi Bruce and T ari We are here 1955 1986 1995 2002 2015 80's Contact geometry with special submanifolds Height function, orthogonal projections, distance square function, etc.

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Contents • Cross-ratio

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Contents • Cross-ratio • Motivation in R 3

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Contents • Cross-ratio • Motivation in R 3 • Surfaces in R 4

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Contents • Cross-ratio • Motivation in R 3 • Surfaces in R 4 • Cross-ratio for surfaces in R 4

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Cross-ratio

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Cross-ratio l 1 l 2 A' B' l 3 A B C' C l 4 D' D

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Cross-ratio l 1 l 2 A' B' l 3 A B C' C l 4 D' D ρ = C − A C − B · D − B D − A

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Cross-ratio l 1 l 2 A' B' l 3 A B C' C l 4 D' D D − A = C ′ − A ′ C ′ − B ′ · D ′ − B ′ ρ = C − A C − B · D − B D ′ − A ′

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Cross-ratio l 1 l 2 A' B' l 3 A B C' C l 4 D' D D − A = C ′ − A ′ C ′ − B ′ · D ′ − B ′ ρ = C − A C − B · D − B D ′ − A ′ if l 2 has infinite ρ = C − A D − A

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Surfaces in R 3 Definition Given a surface S ⊂ R 3 . The Gauss map is given by S 2 N : S → φ x × φ y p �→ N ( p ) = || φ x × φ y || ( p ) , where φ is a parametrization of S .

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Surfaces in R 3 Let p ∈ S and dN p : T p S → T p S . The Gaussian curvature is K = det ( dN p ) . A point p ∈ S is 1. Elliptic, if K > 0; 2. Hyperbolic, if K < 0; 3. Parabolic , is K = 0.

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Surfaces in R 3 Elliptic Parabolic Hyperbolic Gaussian Curvature: K = det( dN p ).

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Surfaces in R 3 Elliptic Parabolic Hyperbolic Asymptotic directions: ∃ 2 directions in hyperbolic region

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Surfaces in R 3 Elliptic Parabolic Field of pairs of directions Hyperbolic Asymptotic directions: ∃ ! direction in parabolic region

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Surfaces in R 3 Elliptic Parabolic Field of pairs of directions Hyperbolic Asymptotic curves

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Surfaces in R 3 Elliptic Parabolic Field of pairs of directions Hyperbolic Asymptotic curves

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Surfaces in R 3 Elliptic Parabolic Field of pairs asymptotic curve of directions Hyperbolic

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Surfaces in R 3 Elliptic Parabolic Field of pairs asymptotic curve of directions Hyperbolic Definition A Cusp of Gauss is a parabolic point at which the single asymptotic direction is tangent to the parabolic curve.

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Surfaces in R 3 Why the name Cusp of Gauss?

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Monge Form and Height Functions We can take the parametrization locally at origin in Monge form φ : R 2 , 0 M ⊂ R 3 → ( x , y ) �→ ( x , y , f ( x , y )) where df (0 , 0) = 0.

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Monge Form and Height Functions We can take the parametrization locally at origin in Monge form φ : R 2 , 0 M ⊂ R 3 → ( x , y ) �→ ( x , y , f ( x , y )) where df (0 , 0) = 0. The contact of M with planes is determined by singularities of H : M × S 2 → R ( x , y , v ) �→ h v = � φ ( x , y ) , v �

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Monge Form and Height Functions We can take the parametrization locally at origin in Monge form φ : R 2 , 0 M ⊂ R 3 → ( x , y ) �→ ( x , y , f ( x , y )) where df (0 , 0) = 0. The contact of M with planes is determined by singularities of H : M × S 2 → R ( x , y , v ) �→ h v = � φ ( x , y ) , v � If v = (0 , 0 , 1) then H v = f ( x , y ) .

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Height Functions Fixed v ∈ S 2 , H v is equivalent to: 1 : x 2 ± y 2 ⇔ p is not parabolic • A ± • A 2 : x 2 + y 3 ⇔ p is parabolic point 3 : x 2 ± y 4 ⇔ p is parabolic and a Cusp of Gauss • A ± A + 1 Elliptic Parabolic A >2 A 3 - Field of pairs asymptotic curve of directions A - 1 Hyperbolic

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Height Functions Figure: A + Figure: A − 1 1 Figure: A + Figure: A 2 3 Figure: A − 3

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Asymptotic Curves (Platonova, 1981) Monge-form ( x , y , f ( x , y )) , the 4-jet of a surface at A 3 can be sent by projective transformation to the normal form j 4 f ( x , y ) = y 2 2 − x 2 y + λ x 4 , λ � = 0 , 1 2

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Asymptotic Curves (Platonova, 1981) Monge-form ( x , y , f ( x , y )) , the 4-jet of a surface at A 3 can be sent by projective transformation to the normal form j 4 f ( x , y ) = y 2 2 − x 2 y + λ x 4 , λ � = 0 , 1 2 A 3 parabolic curve Discriminant: f 2 xy − f xx f yy = 0 (parabolic curve)

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Parabolic Curves (Platonova, 1981) Monge-form ( x , y , f ( x , y )) , the 4-jet of a surface at A 3 can be sent by projective transformation to the normal form j 4 f ( x , y ) = y 2 2 − x 2 y + λ x 4 , λ � = 0 , 1 2 A 3 parabolic curve y = 2(3 λ − 1) x 2 + h . o . t

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Inflection of asymptotic curve (Platonova, 1981) Monge-form ( x , y , f ( x , y )) , the 4-jet of a surface at A 3 can be sent by projective transformation to the normal form j 4 f ( x , y ) = y 2 2 − x 2 y + λ x 4 , λ � = 0 , 1 2 flecnodal A 3 parabolic curve y = 2 λ (4 λ − 1) x 2 + h . o . t

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Inflection of asymptotic curve (Platonova, 1981) Monge-form ( x , y , f ( x , y )) , the 4-jet of a surface at A 3 can be sent by projective transformation to the normal form j 4 f ( x , y ) = y 2 2 − x 2 y + λ x 4 , λ � = 0 , 1 2 flecnodal A 3 parabolic curve y = 2 λ (4 λ − 1) x 2 + h . o . t

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Multi-local curve (Platonova, 1981) Monge-form ( x , y , f ( x , y )) , the 4-jet of a surface at A 3 can be sent by projective transformation to the normal form j 4 f ( x , y ) = y 2 2 − x 2 y + λ x 4 , λ � = 0 , 1 2 v multi-local points

Motivation in R 3 Surfaces in R 4 Cross-ratio for surfaces in R 4 Cross-ratio Multi-local curve (Platonova, 1981) Monge-form ( x , y , f ( x , y )) , the 4-jet of a surface at A 3 can be sent by projective transformation to the normal form j 4 f ( x , y ) = y 2 2 − x 2 y + λ x 4 , λ � = 0 , 1 2 A A conodal ( ) 1 1 flecnodal A 3 parabolic curve y = 2 λ x 2 + h . o . t