Extrinsically flat surfaces of space forms and the
geometric structure on the space of oriented geodesics
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Atsufumi Honda
Tokyo Institute of Technology
Sep 7, 2011
VI International Meeting on Lorentzian Geometry
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Flat v.s Extrinsically Flat
Σ3 = Σ3 (c): a space form (R3 , S3 , H 3 )，
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Def
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A surface f : M 2 → Σ3 is called
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def
B flat : ⇐⇒ K = 0 (K : the sectional curvature of ds2 = f ∗ h, i)，
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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def
B extrinsically flat : ⇐⇒ Kext = 0 (Kext = λ1 λ2 : the Gauss-Kronecker
curvature)．
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2 / 11
Flat v.s Extrinsically Flat
Σ3 = Σ3 (c): a space form (R3 , S3 , H 3 )，
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Def
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A surface f : M 2 → Σ3 is called
.
def
B flat : ⇐⇒ K = 0 (K : the sectional curvature of ds2 = f ∗ h, i)，
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def
B extrinsically flat : ⇐⇒ Kext = 0 (Kext = λ1 λ2 : the Gauss-Kronecker
curvature)．
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Gauss equation:
K = Kext + c.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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2 / 11
Flat v.s Extrinsically Flat
Σ3 = Σ3 (c): a space form (R3 , S3 , H 3 )，
.
Def
..
A surface f : M 2 → Σ3 is called
.
def
B flat : ⇐⇒ K = 0 (K : the sectional curvature of ds2 = f ∗ h, i)，
.
.
def
B extrinsically flat : ⇐⇒ Kext = 0 (Kext = λ1 λ2 : the Gauss-Kronecker
curvature)．
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..
Gauss equation:
K = Kext + c.
Classifications of complete flat & e-flat surfaces:
flat
e-flat
H3
horosphere, tube
∃∞
R3
S3
∃∞
totally geodesic
cylinder
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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2 / 11
Flat v.s Extrinsically Flat
Σ3 = Σ3 (c): a space form (R3 , S3 , H 3 )，
.
Def
..
A surface f : M 2 → Σ3 is called
.
def
B flat : ⇐⇒ K = 0 (K : the sectional curvature of ds2 = f ∗ h, i)，
.
.
def
B extrinsically flat : ⇐⇒ Kext = 0 (Kext = λ1 λ2 : the Gauss-Kronecker
curvature)．
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..
Gauss equation:
K = Kext + c.
Classifications of complete flat & e-flat surfaces:
flat
e-flat
H3
horosphere, tube
∃∞
R3
S3
∃∞
totally geodesic
cylinder
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Fact [Volkov-Vladimirova 0 71, Sasaki 0 73]
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Complete
flat surfaces in H 3 =⇒ horosphere or tube around a geodesic.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Fact [Volkov-Vladimirova 0 71, Sasaki 0 73]
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Complete
flat surfaces in H 3 =⇒ horosphere or tube around a geodesic.
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∃Weierstrass type representation formula [Galvez-Martinez-Milan 0 00]
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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4 / 11
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Fact [Volkov-Vladimirova 0 71, Sasaki 0 73]
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Complete
flat surfaces in H 3 =⇒ horosphere or tube around a geodesic.
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∃Weierstrass type representation formula [Galvez-Martinez-Milan 0 00]
Extension for flat fronts, Osserman-type inequality
[Kokubu-Umehara-Yamada 0 04]
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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4 / 11
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Fact [Volkov-Vladimirova 0 71, Sasaki 0 73]
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Complete
flat surfaces in H 3 =⇒ horosphere or tube around a geodesic.
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∃Weierstrass type representation formula [Galvez-Martinez-Milan 0 00]
Extension for flat fronts, Osserman-type inequality
[Kokubu-Umehara-Yamada 0 04]
Def (wave fronts)
A (wave) front is a C∞ -map f : M 2 → Σ3 such that
• ∃ local unit normal field ν,
• (f , ν) : Up → T1 Σ : immersion.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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4 / 11
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Fact [Volkov-Vladimirova 0 71, Sasaki 0 73]
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Complete
flat surfaces in H 3 =⇒ horosphere or tube around a geodesic.
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∃Weierstrass type representation formula [Galvez-Martinez-Milan 0 00]
Extension for flat fronts, Osserman-type inequality
[Kokubu-Umehara-Yamada 0 04]
Def (wave fronts)
A (wave) front is a C∞ -map f : M 2 → Σ3 such that
• ∃ local unit normal field ν,
0
0
Fact [Hartman-Nirenberg 59, Massey 62]
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Complete
flat surfaces in R3 =⇒ cylinder.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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• (f , ν) : Up → T1 Σ : immersion.
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4 / 11
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Fact [Volkov-Vladimirova 0 71, Sasaki 0 73]
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Complete
flat surfaces in H 3 =⇒ horosphere or tube around a geodesic.
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..
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∃Weierstrass type representation formula [Galvez-Martinez-Milan 0 00]
Extension for flat fronts, Osserman-type inequality
[Kokubu-Umehara-Yamada 0 04]
Def (wave fronts)
A (wave) front is a C∞ -map f : M 2 → Σ3 such that
• ∃ local unit normal field ν,
0
0
Fact [Hartman-Nirenberg 59, Massey 62]
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Complete
flat surfaces in R3 =⇒ cylinder.
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• (f , ν) : Up → T1 Σ : immersion.
Classification of complete flat fronts,
estimation for #{singularities other than cuspidal edge}
[Murata-Umehara 0 09]
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Fact [O’Neill-Stiel 0 63]
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Compact
extrinsically flat surfaces in S3 =⇒ totally geodesic.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Fact [O’Neill-Stiel 0 63]
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Compact
extrinsically flat surfaces in S3 =⇒ totally geodesic.
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B Compact extrinsically flat fronts in S3 =⇒ ∃ many!
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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5 / 11
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Fact [O’Neill-Stiel 0 63]
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Compact
extrinsically flat surfaces in S3 =⇒ totally geodesic.
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B Compact extrinsically flat fronts in S3 =⇒ ∃ many!
Figure: Compact e-flat fronts in S3
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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5 / 11
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Fact [O’Neill-Stiel 0 63]
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Compact
extrinsically flat surfaces in S3 =⇒ totally geodesic.
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B Compact extrinsically flat fronts in S3 =⇒ ∃ many!
Figure: Compact e-flat fronts in S3
Q. How many? Classify compact extrinsically flat fronts in S3 .
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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5 / 11
Main results
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Main theorem
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Closed extrinsically flat fronts in S3
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corresp.
←→ pairs of closed curve (γ1 (s), γ2 (s)) in S2 s.t.
(1) s is the arc-length parameter,
. (3) L1 /L2 is rational
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(κ1 , κ2 : the curvature functions of γ1 , γ2 ),
(L1 , L2 : the lengths).
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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(2) κ1 (s) > −κ2 (s)
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6 / 11
Main results
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Main theorem
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Closed extrinsically flat fronts in S3
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corresp.
←→ pairs of closed curve (γ1 (s), γ2 (s)) in S2 s.t.
(1) s is the arc-length parameter,
. (3) L1 /L2 is rational
..
(κ1 , κ2 : the curvature functions of γ1 , γ2 ),
(L1 , L2 : the lengths).
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(2) κ1 (s) > −κ2 (s)
B Construction & proof:
(1) Closed e-flat fronts =⇒ developable (e-flat & ruled).
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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6 / 11
Main results
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Main theorem
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Closed extrinsically flat fronts in S3
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corresp.
←→ pairs of closed curve (γ1 (s), γ2 (s)) in S2 s.t.
(1) s is the arc-length parameter,
. (3) L1 /L2 is rational
..
(κ1 , κ2 : the curvature functions of γ1 , γ2 ),
(L1 , L2 : the lengths).
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(2) κ1 (s) > −κ2 (s)
B Construction & proof:
(1) Closed e-flat fronts =⇒ developable (e-flat & ruled).
corresp.
(2) Ruled surfaces ←→ curves in L(S3 )
(L(S3 ) := {ori. geod. in S3 }),
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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6 / 11
Main results
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Main theorem
..
Closed extrinsically flat fronts in S3
.
corresp.
←→ pairs of closed curve (γ1 (s), γ2 (s)) in S2 s.t.
(1) s is the arc-length parameter,
. (3) L1 /L2 is rational
..
(κ1 , κ2 : the curvature functions of γ1 , γ2 ),
(L1 , L2 : the lengths).
.
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(2) κ1 (s) > −κ2 (s)
B Construction & proof:
(1) Closed e-flat fronts =⇒ developable (e-flat & ruled).
corresp.
(2) Ruled surfaces ←→ curves in L(S3 )
corresp.
(L(S3 ) := {ori. geod. in S3 }),
Developables ←→ null curves in (L(S3 ), Gmini twistor )
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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6 / 11
Main results
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Main theorem
..
Closed extrinsically flat fronts in S3
.
corresp.
←→ pairs of closed curve (γ1 (s), γ2 (s)) in S2 s.t.
(1) s is the arc-length parameter,
. (3) L1 /L2 is rational
..
(κ1 , κ2 : the curvature functions of γ1 , γ2 ),
(L1 , L2 : the lengths).
.
.
(2) κ1 (s) > −κ2 (s)
B Construction & proof:
(1) Closed e-flat fronts =⇒ developable (e-flat & ruled).
corresp.
(2) Ruled surfaces ←→ curves in L(S3 )
corresp.
(L(S3 ) := {ori. geod. in S3 }),
Developables ←→ null curves in (L(S3 ), Gmini twistor ) = S2 × S2 .
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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6 / 11
Main results
.
Main theorem
..
Closed extrinsically flat fronts in S3
.
corresp.
←→ pairs of closed curve (γ1 (s), γ2 (s)) in S2 s.t.
(1) s is the arc-length parameter,
. (3) L1 /L2 is rational
..
(κ1 , κ2 : the curvature functions of γ1 , γ2 ),
(L1 , L2 : the lengths).
.
.
(2) κ1 (s) > −κ2 (s)
B Construction & proof:
(1) Closed e-flat fronts =⇒ developable (e-flat & ruled).
corresp.
(2) Ruled surfaces ←→ curves in L(S3 )
corresp.
(L(S3 ) := {ori. geod. in S3 }),
Developables ←→ null curves in (L(S3 ), Gmini twistor ) = S2 × S2 .
I Applications :
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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6 / 11
Main results
.
Main theorem
..
Closed extrinsically flat fronts in S3
.
corresp.
←→ pairs of closed curve (γ1 (s), γ2 (s)) in S2 s.t.
(1) s is the arc-length parameter,
. (3) L1 /L2 is rational
..
(κ1 , κ2 : the curvature functions of γ1 , γ2 ),
(L1 , L2 : the lengths).
.
.
(2) κ1 (s) > −κ2 (s)
B Construction & proof:
(1) Closed e-flat fronts =⇒ developable (e-flat & ruled).
corresp.
(2) Ruled surfaces ←→ curves in L(S3 )
corresp.
(L(S3 ) := {ori. geod. in S3 }),
Developables ←→ null curves in (L(S3 ), Gmini twistor ) = S2 × S2 .
I Applications :
Duality for e-flat fronts.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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6 / 11
Main results
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Main theorem
..
Closed extrinsically flat fronts in S3
.
corresp.
←→ pairs of closed curve (γ1 (s), γ2 (s)) in S2 s.t.
(1) s is the arc-length parameter,
. (3) L1 /L2 is rational
..
(κ1 , κ2 : the curvature functions of γ1 , γ2 ),
(L1 , L2 : the lengths).
.
.
(2) κ1 (s) > −κ2 (s)
B Construction & proof:
(1) Closed e-flat fronts =⇒ developable (e-flat & ruled).
corresp.
(2) Ruled surfaces ←→ curves in L(S3 )
corresp.
(L(S3 ) := {ori. geod. in S3 }),
Developables ←→ null curves in (L(S3 ), Gmini twistor ) = S2 × S2 .
I Applications :
Duality for e-flat fronts.
Caustics of e-flat fronts.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Applications
I Duality for e-flat fronts:
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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7 / 11
Applications
I Duality for e-flat fronts:
Duality for geodesics in S3 −→ Duality for ruled surfaces in S3 .
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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7 / 11
Applications
I Duality for e-flat fronts:
Duality for geodesics in S3 −→ Duality for ruled surfaces in S3 .
Dual of e-flat ruled surfaces =⇒ e-flat.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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7 / 11
Applications
I Duality for e-flat fronts:
Duality for geodesics in S3 −→ Duality for ruled surfaces in S3 .
Dual of e-flat ruled surfaces =⇒ e-flat.
Thm2 Classification of self-dual e-flat fronts.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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7 / 11
Applications
I Duality for e-flat fronts:
Duality for geodesics in S3 −→ Duality for ruled surfaces in S3 .
Dual of e-flat ruled surfaces =⇒ e-flat.
Thm2 Classification of self-dual e-flat fronts.
I Caustics of e-flat fronts:
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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7 / 11
Applications
I Duality for e-flat fronts:
Duality for geodesics in S3 −→ Duality for ruled surfaces in S3 .
Dual of e-flat ruled surfaces =⇒ e-flat.
Thm2 Classification of self-dual e-flat fronts.
I Caustics of e-flat fronts:
Caustic (focal surface) : the singular locus of parallel surfaces.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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7 / 11
Applications
I Duality for e-flat fronts:
Duality for geodesics in S3 −→ Duality for ruled surfaces in S3 .
Dual of e-flat ruled surfaces =⇒ e-flat.
Thm2 Classification of self-dual e-flat fronts.
I Caustics of e-flat fronts:
Caustic (focal surface) : the singular locus of parallel surfaces.
[Fact] Caustics of flat fronts are also flat.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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7 / 11
Applications
I Duality for e-flat fronts:
Duality for geodesics in S3 −→ Duality for ruled surfaces in S3 .
Dual of e-flat ruled surfaces =⇒ e-flat.
Thm2 Classification of self-dual e-flat fronts.
I Caustics of e-flat fronts:
Caustic (focal surface) : the singular locus of parallel surfaces.
[Fact] Caustics of flat fronts are also flat.
Caustics of e-flat fronts are also e-flat.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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7 / 11
Applications
I Duality for e-flat fronts:
Duality for geodesics in S3 −→ Duality for ruled surfaces in S3 .
Dual of e-flat ruled surfaces =⇒ e-flat.
Thm2 Classification of self-dual e-flat fronts.
I Caustics of e-flat fronts:
Caustic (focal surface) : the singular locus of parallel surfaces.
[Fact] Caustics of flat fronts are also flat.
Caustics of e-flat fronts are also e-flat.
The inverse problem for caustics:
For a given e-flat front f , find the e-flat front X s.t.
CX = f .
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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7 / 11
Applications
I Duality for e-flat fronts:
Duality for geodesics in S3 −→ Duality for ruled surfaces in S3 .
Dual of e-flat ruled surfaces =⇒ e-flat.
Thm2 Classification of self-dual e-flat fronts.
I Caustics of e-flat fronts:
Caustic (focal surface) : the singular locus of parallel surfaces.
[Fact] Caustics of flat fronts are also flat.
Caustics of e-flat fronts are also e-flat.
The inverse problem for caustics:
For a given e-flat front f , find the e-flat front X s.t.
CX = f .
Thm3 If f : totally geodesic =⇒ @ solution X to the inverse problem.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Examples
Figure: (γ1 , γ2 ) : great circles in
S2 .
Figure: (γ1 , γ2 ) : small circles in
S2 .
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Conelike
singularities.
Dual.
Swallow tails.
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Anti-de Sitter space
B Representation formula for timelike e-flat surfaces in H13 .
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Anti-de Sitter space
B Representation formula for timelike e-flat surfaces in H13 .
B H13 : AdS 3-space (the cplt. simp-conn. Lorltz mfd of sect curv −1)．
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Anti-de Sitter space
B Representation formula for timelike e-flat surfaces in H13 .
B H13 : AdS 3-space (the cplt. simp-conn. Lorltz mfd of sect curv −1)．
B L− (H13 ) := { timelike geodesics in H13 } = H 2 × H 2 ．
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Anti-de Sitter space
B Representation formula for timelike e-flat surfaces in H13 .
B H13 : AdS 3-space (the cplt. simp-conn. Lorltz mfd of sect curv −1)．
B L− (H13 ) := { timelike geodesics in H13 } = H 2 × H 2 ．
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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Thank you for your attention!
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Atsufumi Honda (Tokyo Tech.)
Extrinsically Flat Surfaces.
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Sep 7, 2011
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