Bisectors and foliations in the complex hyperbolic space Maciej Czarnecki Uniwersytet � L´ odzki, � L´ od´ z, Poland Symmetry and shape Universidade de Santiago de Compostela, Spain October 29, 2019 Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Summary 1 Bisectors in complex hyperbolic spaces 2 Complex cross–ratio and Goldman invariant 3 Separating bisectors 4 Representation in de Sitter space Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Summary 1 Bisectors in complex hyperbolic spaces 2 Complex cross–ratio and Goldman invariant 3 Separating bisectors 4 Representation in de Sitter space Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Summary 1 Bisectors in complex hyperbolic spaces 2 Complex cross–ratio and Goldman invariant 3 Separating bisectors 4 Representation in de Sitter space Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Summary 1 Bisectors in complex hyperbolic spaces 2 Complex cross–ratio and Goldman invariant 3 Separating bisectors 4 Representation in de Sitter space Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Summary 1 Bisectors in complex hyperbolic spaces 2 Complex cross–ratio and Goldman invariant 3 Separating bisectors 4 Representation in de Sitter space Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Complex hyperbolic distance Definition For the Hermitian form � X | Y � = X 1 Y 1 + . . . + X n Y n − X n +1 Y n +1 in C n +1 we define n–dimensional complex hyperbolic space as projectivization of negative vectors i.e. C H n = X ∈ C n +1 | � X | X � < 0 � � / C ∗ and its ideal boundary C H n ( ∞ ) as projectivization of null vectors. The Bergman metric makes C H n an Hadamard manifold of sectional curvature between − 1 / 4 and − 1 and the distance given by cosh 2 d ( x , y ) = � X | Y �� Y | X � � X | X �� Y | Y � . 2 Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Complex hyperbolic distance Definition For the Hermitian form � X | Y � = X 1 Y 1 + . . . + X n Y n − X n +1 Y n +1 in C n +1 we define n–dimensional complex hyperbolic space as projectivization of negative vectors i.e. C H n = X ∈ C n +1 | � X | X � < 0 � � / C ∗ and its ideal boundary C H n ( ∞ ) as projectivization of null vectors. The Bergman metric makes C H n an Hadamard manifold of sectional curvature between − 1 / 4 and − 1 and the distance given by cosh 2 d ( x , y ) = � X | Y �� Y | X � � X | X �� Y | Y � . 2 Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Complex geodesics and complex hyperplanes A complex geodesic is the projectivization of a vector space in C n +1 spanned by two linearly indpent negative vectors. It is isometric to real hyperbolic plane R H 2 . A complex hyperplane is the projectivization of a vector space in C n +1 spanned by n linearly indpent negative vectors. It is isometric to C H n − 1 and orthogonal to a unit positive vector (its polar vector ) Proposition Let H 1 and H 2 be complex hyperplanes in C H n with polar vectors C 1 and C 2 . Then 1 H 1 ∩ H 2 = ∅ iff |� C 1 | C 2 �| > 1 . 2 ∠ ( H 1 , H 2 ) = α iff |� C 1 | C 2 �| = cos α . Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Complex geodesics and complex hyperplanes A complex geodesic is the projectivization of a vector space in C n +1 spanned by two linearly indpent negative vectors. It is isometric to real hyperbolic plane R H 2 . A complex hyperplane is the projectivization of a vector space in C n +1 spanned by n linearly indpent negative vectors. It is isometric to C H n − 1 and orthogonal to a unit positive vector (its polar vector ) Proposition Let H 1 and H 2 be complex hyperplanes in C H n with polar vectors C 1 and C 2 . Then 1 H 1 ∩ H 2 = ∅ iff |� C 1 | C 2 �| > 1 . 2 ∠ ( H 1 , H 2 ) = α iff |� C 1 | C 2 �| = cos α . Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Complex geodesics and complex hyperplanes A complex geodesic is the projectivization of a vector space in C n +1 spanned by two linearly indpent negative vectors. It is isometric to real hyperbolic plane R H 2 . A complex hyperplane is the projectivization of a vector space in C n +1 spanned by n linearly indpent negative vectors. It is isometric to C H n − 1 and orthogonal to a unit positive vector (its polar vector ) Proposition Let H 1 and H 2 be complex hyperplanes in C H n with polar vectors C 1 and C 2 . Then 1 H 1 ∩ H 2 = ∅ iff |� C 1 | C 2 �| > 1 . 2 ∠ ( H 1 , H 2 ) = α iff |� C 1 | C 2 �| = cos α . Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Bisectors Definition For z 1 , z 2 ∈ C H n we define a bisector as an equidistant from z 1 and z 2 E ( z 1 , z 2 ) = { z | d ( z , z 1 ) = d ( z , z 2 ) } . Bisectors are in one-to-one correspondence with pairs of points on the ideal boundary C H n ( ∞ ). These points (called vertices of bisector) are ends of the unique geodesic line through z 1 and z 2 . For the bisector E of vertices p and q we call the geodesic line σ a s pine while the complex geodesic Σ = span C ( p , q ) ∩ C H n ≃ C H 1 ≃ R H 2 a complex spine . Observe that E ∩ Σ = σ . Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Bisectors Definition For z 1 , z 2 ∈ C H n we define a bisector as an equidistant from z 1 and z 2 E ( z 1 , z 2 ) = { z | d ( z , z 1 ) = d ( z , z 2 ) } . Bisectors are in one-to-one correspondence with pairs of points on the ideal boundary C H n ( ∞ ). These points (called vertices of bisector) are ends of the unique geodesic line through z 1 and z 2 . For the bisector E of vertices p and q we call the geodesic line σ a s pine while the complex geodesic Σ = span C ( p , q ) ∩ C H n ≃ C H 1 ≃ R H 2 a complex spine . Observe that E ∩ Σ = σ . Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Bisectors Definition For z 1 , z 2 ∈ C H n we define a bisector as an equidistant from z 1 and z 2 E ( z 1 , z 2 ) = { z | d ( z , z 1 ) = d ( z , z 2 ) } . Bisectors are in one-to-one correspondence with pairs of points on the ideal boundary C H n ( ∞ ). These points (called vertices of bisector) are ends of the unique geodesic line through z 1 and z 2 . For the bisector E of vertices p and q we call the geodesic line σ a s pine while the complex geodesic Σ = span C ( p , q ) ∩ C H n ≃ C H 1 ≃ R H 2 a complex spine . Observe that E ∩ Σ = σ . Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Properties of bisectors 1 A bisector is a real analytic fibration over its spine with respect to the orthogonal projection onto the complex spine z ∈ σ Π − 1 E = � Σ ( z ) ( slice decomposition ). 2 For z ∈ C H n the bisector E is equidistant from z iff z ∈ Σ \ σ . 3 A bisector is a real hypersurface which is Hadamard and even in C H 2 it has 3 distinct principal curvatures: − 1, − 1 / 4 and some between − 1 / 2 and − 1 / 4. 4 Every two bisectors are congruent Observe that in case of R H n all these properties trivialize — bisectors are totally geodesic. Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Properties of bisectors 1 A bisector is a real analytic fibration over its spine with respect to the orthogonal projection onto the complex spine z ∈ σ Π − 1 E = � Σ ( z ) ( slice decomposition ). 2 For z ∈ C H n the bisector E is equidistant from z iff z ∈ Σ \ σ . 3 A bisector is a real hypersurface which is Hadamard and even in C H 2 it has 3 distinct principal curvatures: − 1, − 1 / 4 and some between − 1 / 2 and − 1 / 4. 4 Every two bisectors are congruent Observe that in case of R H n all these properties trivialize — bisectors are totally geodesic. Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Properties of bisectors 1 A bisector is a real analytic fibration over its spine with respect to the orthogonal projection onto the complex spine z ∈ σ Π − 1 E = � Σ ( z ) ( slice decomposition ). 2 For z ∈ C H n the bisector E is equidistant from z iff z ∈ Σ \ σ . 3 A bisector is a real hypersurface which is Hadamard and even in C H 2 it has 3 distinct principal curvatures: − 1, − 1 / 4 and some between − 1 / 2 and − 1 / 4. 4 Every two bisectors are congruent Observe that in case of R H n all these properties trivialize — bisectors are totally geodesic. Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
Properties of bisectors 1 A bisector is a real analytic fibration over its spine with respect to the orthogonal projection onto the complex spine z ∈ σ Π − 1 E = � Σ ( z ) ( slice decomposition ). 2 For z ∈ C H n the bisector E is equidistant from z iff z ∈ Σ \ σ . 3 A bisector is a real hypersurface which is Hadamard and even in C H 2 it has 3 distinct principal curvatures: − 1, − 1 / 4 and some between − 1 / 2 and − 1 / 4. 4 Every two bisectors are congruent Observe that in case of R H n all these properties trivialize — bisectors are totally geodesic. Maciej Czarnecki Bisectors and foliations in the complex hyperbolic space
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