Lagrangian submanifolds in the nearly Kähler...
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Almost complex surfaces Lagrangian submanifolds in the nearly K¨ ahler S 3 × S 3 Luc Vrancken Based on joint work Bart Dioos and J Xianfeng Wang (Nankai University) Zlatibor September, 2016
Transcript of Lagrangian submanifolds in the nearly Kähler...
Almost complex surfaces
Lagrangian submanifolds in the nearlyKahler S3 × S3
Luc Vrancken
Based on joint work Bart Dioos and J Xianfeng Wang (NankaiUniversity)
ZlatiborSeptember, 2016
Almost complex surfaces
Abstract
We will discuss lagrangian submanifolds in S3 × S3. The main topicswe discuss are the following:
1 Nearly Kahler manifolds
2 The construction of the nearly Kaehler structure on S3 × S3
3 Its relation with the Euclidean induced structure
4 Basic properties of Lagrangian submanifolds
5 Elementary examples of Lagrangian submanifolds
6 Totally geodesic Lagrangian submanifolds
7 Lagrangian submanifolds with constant sectional curvature
Almost complex surfaces
Abstract
We will discuss lagrangian submanifolds in S3 × S3. The main topicswe discuss are the following:
1 Nearly Kahler manifolds
2 The construction of the nearly Kaehler structure on S3 × S3
3 Its relation with the Euclidean induced structure
4 Basic properties of Lagrangian submanifolds
5 Elementary examples of Lagrangian submanifolds
6 Totally geodesic Lagrangian submanifolds
7 Lagrangian submanifolds with constant sectional curvature
Almost complex surfaces
Abstract
We will discuss lagrangian submanifolds in S3 × S3. The main topicswe discuss are the following:
1 Nearly Kahler manifolds
2 The construction of the nearly Kaehler structure on S3 × S3
3 Its relation with the Euclidean induced structure
4 Basic properties of Lagrangian submanifolds
5 Elementary examples of Lagrangian submanifolds
6 Totally geodesic Lagrangian submanifolds
7 Lagrangian submanifolds with constant sectional curvature
Almost complex surfaces
Abstract
We will discuss lagrangian submanifolds in S3 × S3. The main topicswe discuss are the following:
1 Nearly Kahler manifolds
2 The construction of the nearly Kaehler structure on S3 × S3
3 Its relation with the Euclidean induced structure
4 Basic properties of Lagrangian submanifolds
5 Elementary examples of Lagrangian submanifolds
6 Totally geodesic Lagrangian submanifolds
7 Lagrangian submanifolds with constant sectional curvature
Almost complex surfaces
Abstract
We will discuss lagrangian submanifolds in S3 × S3. The main topicswe discuss are the following:
1 Nearly Kahler manifolds
2 The construction of the nearly Kaehler structure on S3 × S3
3 Its relation with the Euclidean induced structure
4 Basic properties of Lagrangian submanifolds
5 Elementary examples of Lagrangian submanifolds
6 Totally geodesic Lagrangian submanifolds
7 Lagrangian submanifolds with constant sectional curvature
Almost complex surfaces
Abstract
We will discuss lagrangian submanifolds in S3 × S3. The main topicswe discuss are the following:
1 Nearly Kahler manifolds
2 The construction of the nearly Kaehler structure on S3 × S3
3 Its relation with the Euclidean induced structure
4 Basic properties of Lagrangian submanifolds
5 Elementary examples of Lagrangian submanifolds
6 Totally geodesic Lagrangian submanifolds
7 Lagrangian submanifolds with constant sectional curvature
Almost complex surfaces
Abstract
We will discuss lagrangian submanifolds in S3 × S3. The main topicswe discuss are the following:
1 Nearly Kahler manifolds
2 The construction of the nearly Kaehler structure on S3 × S3
3 Its relation with the Euclidean induced structure
4 Basic properties of Lagrangian submanifolds
5 Elementary examples of Lagrangian submanifolds
6 Totally geodesic Lagrangian submanifolds
7 Lagrangian submanifolds with constant sectional curvature
Almost complex surfaces
Abstract
We will discuss lagrangian submanifolds in S3 × S3. The main topicswe discuss are the following:
1 Nearly Kahler manifolds
2 The construction of the nearly Kaehler structure on S3 × S3
3 Its relation with the Euclidean induced structure
4 Basic properties of Lagrangian submanifolds
5 Elementary examples of Lagrangian submanifolds
6 Totally geodesic Lagrangian submanifolds
7 Lagrangian submanifolds with constant sectional curvature
Almost complex surfaces
Nearly Kahler manifolds
Nearly Kahler manifolds
Almost complex surfaces
Nearly Kahler manifolds
Nearly Kahler manifoldsDefinitions
An almost Hermitian manifold (M, g , J) is a manifold M withmetric g and almost complex structure J (endomorphism s.t.J2 = −Id) satisfying
g(JX , JY ) = g(X ,Y ), X ,Y ∈ TM.
A nearly Kahler manifold is an almost Hermitian manifold (M, g , J)with the extra assumption that ∇J is skew-symmetric:
(∇X J)Y + (∇Y J)X = 0, X ,Y ∈ TM.
First example introduced by Fukami and Ishihara in 1955 andsystematically studied by A. Gray in several papers starting from 1970.
Almost complex surfaces
Nearly Kahler manifolds
Nearly Kahler manifoldsDefinitions
An almost Hermitian manifold (M, g , J) is a manifold M withmetric g and almost complex structure J (endomorphism s.t.J2 = −Id) satisfying
g(JX , JY ) = g(X ,Y ), X ,Y ∈ TM.
A nearly Kahler manifold is an almost Hermitian manifold (M, g , J)with the extra assumption that ∇J is skew-symmetric:
(∇X J)Y + (∇Y J)X = 0, X ,Y ∈ TM.
First example introduced by Fukami and Ishihara in 1955 andsystematically studied by A. Gray in several papers starting from 1970.
Almost complex surfaces
Nearly Kahler manifolds
Nearly Kahler manifoldsIdentities
For convenience, we will write G (X ,Y ) = (∇X J)Y .
Some identities:
G (X ,Y ) + G (Y ,X ) = 0,
G (X , JY ) + JG (X ,Y ) = 0,
g(G (X ,Y ),Z ) and g(G (X ,Y ), JZ ) are totally anti symmetric.
∇J = 0.
Here ∇XY = ∇XY − 12JG (X ,Y ) is the canonical Hermitian
connection.
Almost complex surfaces
Nearly Kahler manifolds
For nearly Kahler manifolds the 6-dimensional case is particularlyinteresting, as
1 by the structure theorems of Nagy, they serve as building blocks forarbitrary nearly Kahler manifolds
2 by a result of Grunewald, provided the nearly Kahler manifold issimply connected, there is a bijective correspondence between nearlyKahler structures and unit real Killing spinors
Almost complex surfaces
Nearly Kahler manifolds
For nearly Kahler manifolds the 6-dimensional case is particularlyinteresting, as
1 by the structure theorems of Nagy, they serve as building blocks forarbitrary nearly Kahler manifolds
2 by a result of Grunewald, provided the nearly Kahler manifold issimply connected, there is a bijective correspondence between nearlyKahler structures and unit real Killing spinors
Almost complex surfaces
Nearly Kahler manifolds
Recently it has been shown by Butruille that the only homogeneous6-dimensional nearly Kahler manifolds are
1 the nearly Kahler 6-sphere,
2 S3 × S3,
3 the projective space CP3 (but not with the usual metric andcomplex structure)
4 M = SU(3)/U(1)× U(1), the space of flags of C3.
All these spaces are compact.
Almost complex surfaces
Nearly Kahler manifolds
Recently it has been shown by Butruille that the only homogeneous6-dimensional nearly Kahler manifolds are
1 the nearly Kahler 6-sphere,
2 S3 × S3,
3 the projective space CP3 (but not with the usual metric andcomplex structure)
4 M = SU(3)/U(1)× U(1), the space of flags of C3.
All these spaces are compact.
Almost complex surfaces
Nearly Kahler manifolds
Recently it has been shown by Butruille that the only homogeneous6-dimensional nearly Kahler manifolds are
1 the nearly Kahler 6-sphere,
2 S3 × S3,
3 the projective space CP3 (but not with the usual metric andcomplex structure)
4 M = SU(3)/U(1)× U(1), the space of flags of C3.
All these spaces are compact.
Almost complex surfaces
Nearly Kahler manifolds
Recently it has been shown by Butruille that the only homogeneous6-dimensional nearly Kahler manifolds are
1 the nearly Kahler 6-sphere,
2 S3 × S3,
3 the projective space CP3 (but not with the usual metric andcomplex structure)
4 M = SU(3)/U(1)× U(1), the space of flags of C3.
All these spaces are compact.
Almost complex surfaces
Nearly Kahler manifolds
Recently it has been shown by Butruille that the only homogeneous6-dimensional nearly Kahler manifolds are
1 the nearly Kahler 6-sphere,
2 S3 × S3,
3 the projective space CP3 (but not with the usual metric andcomplex structure)
4 M = SU(3)/U(1)× U(1), the space of flags of C3.
All these spaces are compact.
Almost complex surfaces
The nearly Kahler S3 × S3
The nearly Kahler S3 × S3
Almost complex surfaces
The nearly Kahler S3 × S3
The nearly Kahler S3 × S3
The nearly Kahler structure
LetZ (p, q) = (pU(p, q), qV (p, q)),
be a tangent vector at the point (p, q). Then U(p, q) and V (p, q) areimaginary quaternions.
The almost complex structure J on S3 × S3 is defined by
JZ(p,q) =1√3
(p(2V − U), q(−2U + V )) .
If 〈· , ·〉 is the product metric on S3 × S3, the metric g is given by
g(Z ,Z ′) =1
2(〈Z ,Z ′〉+ 〈JZ , JZ ′〉)
=4
3(〈U,U ′〉+ 〈V ,V ′〉)
− 2
3(〈U,V ′〉+ 〈U ′,V 〉) .
Almost complex surfaces
The nearly Kahler S3 × S3
The nearly Kahler S3 × S3
The nearly Kahler structure
LetZ (p, q) = (pU(p, q), qV (p, q)),
be a tangent vector at the point (p, q). Then U(p, q) and V (p, q) areimaginary quaternions.
The almost complex structure J on S3 × S3 is defined by
JZ(p,q) =1√3
(p(2V − U), q(−2U + V )) .
If 〈· , ·〉 is the product metric on S3 × S3, the metric g is given by
g(Z ,Z ′) =1
2(〈Z ,Z ′〉+ 〈JZ , JZ ′〉)
=4
3(〈U,U ′〉+ 〈V ,V ′〉)
− 2
3(〈U,V ′〉+ 〈U ′,V 〉) .
Almost complex surfaces
The nearly Kahler S3 × S3
The nearly Kahler S3 × S3
The nearly Kahler structure
LetZ (p, q) = (pU(p, q), qV (p, q)),
be a tangent vector at the point (p, q). Then U(p, q) and V (p, q) areimaginary quaternions.
The almost complex structure J on S3 × S3 is defined by
JZ(p,q) =1√3
(p(2V − U), q(−2U + V )) .
If 〈· , ·〉 is the product metric on S3 × S3, the metric g is given by
g(Z ,Z ′) =1
2(〈Z ,Z ′〉+ 〈JZ , JZ ′〉)
=4
3(〈U,U ′〉+ 〈V ,V ′〉)
− 2
3(〈U,V ′〉+ 〈U ′,V 〉) .
Almost complex surfaces
The nearly Kahler S3 × S3
In S3 × S3 we have that the tensor G has the following additionalproperties:
g(G (X ,Y ),G (Z ,W )
)=
1
3
(g(X ,Z )g(Y ,W )− g(X ,W )g(Y ,Z )
+ g(JX ,Z )g(JW ,Y )− g(JX ,W )g(JZ ,Y )),
G(X ,G (Y ,Z )
)=
1
3
(g(X ,Z )Y − g(X ,Y )Z
+ g(JX ,Z )JY − g(JX ,Y )JZ),
(∇G )(X ,Y ,Z ) =1
3(g(X ,Z )JY − g(X ,Y )JZ − g(JY ,Z )X ).
Almost complex surfaces
The nearly Kahler S3 × S3
The nearly Kahler S3 × S3
Isometries
For unit quaternions a, b and c the map
F (p, q) = (apc−1, bqc−1)
is an isometry.
Almost complex surfaces
The nearly Kahler S3 × S3
The nearly Kahler S3 × S3
An almost product structure P
We define the almost product structure P as
P(Z )(p,q) = (pV , qU).
Properties:
P2 = Id,
PJ = −JP,g(PZ ,PZ ′) = g(Z ,Z ′),
P is preserved by isometries,
PG (X ,Y ) + G (PX ,PY )
Note that the usual product structure
Q(Z )(p,q) = (−pU, qV ).
is not compatible with the almost complex metric.
Almost complex surfaces
The nearly Kahler S3 × S3
The nearly Kahler S3 × S3
An almost product structure P
We define the almost product structure P as
P(Z )(p,q) = (pV , qU).
Properties:
P2 = Id,
PJ = −JP,g(PZ ,PZ ′) = g(Z ,Z ′),
P is preserved by isometries,
PG (X ,Y ) + G (PX ,PY )
Note that the usual product structure
Q(Z )(p,q) = (−pU, qV ).
is not compatible with the almost complex metric.
Almost complex surfaces
The nearly Kahler S3 × S3
The nearly Kahler S3 × S3
An almost product structure P
We define the almost product structure P as
P(Z )(p,q) = (pV , qU).
Properties:
P2 = Id,
PJ = −JP,g(PZ ,PZ ′) = g(Z ,Z ′),
P is preserved by isometries,
PG (X ,Y ) + G (PX ,PY )
Note that the usual product structure
Q(Z )(p,q) = (−pU, qV ).
is not compatible with the almost complex metric.
Almost complex surfaces
The nearly Kahler S3 × S3
The nearly Kahler S3 × S3
Curvature tensor
The Riemann curvature tensor of S3 × S3 is
R(U,V )W =5
12
(g(V ,W )U − g(U,W )V
)+
1
12
(g(JV ,W )JU − g(JU,W )JV − 2g(JU,V )JW
)+
1
3
(g(PV ,W )PU − g(PU,W )PV
+ g(JPV ,W )JPU − g(JPU,W )JPV).
(∇ZP)Z ′ = 12J(G (Z ,PZ ′) + PG (Z ,Z ′))
Almost complex surfaces
Relations with the Euclidean induced structure
Relations with the Euclidean induced structure
Almost complex surfaces
Relations with the Euclidean induced structure
Remark that the Levi Civita connections ∇ of g and ∇ of the usualmetric are related by
∇ZZ′ = ∇ZZ
′ +1
2(JG (Z ,PZ ′) + JG (Z ′,PZ )).
QZ = 1√3
(2PJZ − JZ ).
and
〈Z ,Z ′〉 = g(Z ,Z ′) + 12g(Z ,PZ ′)
〈Z ,QZ ′〉 =
√3
2g(Z ,PJZ ′)
Note that the ∇ is given by
DZZ′ = ∇ZZ
′ − 12 〈Z ,QZ
′〉(−p, q)− 12 〈Z ,Z
′〉(p, q).
Almost complex surfaces
Relations with the Euclidean induced structure
Remark that the Levi Civita connections ∇ of g and ∇ of the usualmetric are related by
∇ZZ′ = ∇ZZ
′ +1
2(JG (Z ,PZ ′) + JG (Z ′,PZ )).
QZ = 1√3
(2PJZ − JZ ).
and
〈Z ,Z ′〉 = g(Z ,Z ′) + 12g(Z ,PZ ′)
〈Z ,QZ ′〉 =
√3
2g(Z ,PJZ ′)
Note that the ∇ is given by
DZZ′ = ∇ZZ
′ − 12 〈Z ,QZ
′〉(−p, q)− 12 〈Z ,Z
′〉(p, q).
Almost complex surfaces
Relations with the Euclidean induced structure
Remark that the Levi Civita connections ∇ of g and ∇ of the usualmetric are related by
∇ZZ′ = ∇ZZ
′ +1
2(JG (Z ,PZ ′) + JG (Z ′,PZ )).
QZ = 1√3
(2PJZ − JZ ).
and
〈Z ,Z ′〉 = g(Z ,Z ′) + 12g(Z ,PZ ′)
〈Z ,QZ ′〉 =
√3
2g(Z ,PJZ ′)
Note that the ∇ is given by
DZZ′ = ∇ZZ
′ − 12 〈Z ,QZ
′〉(−p, q)− 12 〈Z ,Z
′〉(p, q).
Almost complex surfaces
Relations with the Euclidean induced structure
Remark that the Levi Civita connections ∇ of g and ∇ of the usualmetric are related by
∇ZZ′ = ∇ZZ
′ +1
2(JG (Z ,PZ ′) + JG (Z ′,PZ )).
QZ = 1√3
(2PJZ − JZ ).
and
〈Z ,Z ′〉 = g(Z ,Z ′) + 12g(Z ,PZ ′)
〈Z ,QZ ′〉 =
√3
2g(Z ,PJZ ′)
Note that the ∇ is given by
DZZ′ = ∇ZZ
′ − 12 〈Z ,QZ
′〉(−p, q)− 12 〈Z ,Z
′〉(p, q).
Almost complex surfaces
Basic properties of Lagrangian submanifolds
Basic properties of Lagrangian submanifolds
Almost complex surfaces
Basic properties of Lagrangian submanifolds
A 3-dimensional manifold M3 in S3 × S3 is called Lagrnagian if J mapsthe tangent space into the normal space.
In view of the dimension this also implies that J of a normal vector istangent, i.e. J is an isomorphism between the tangent and the normalspace.
Theorem (Schafer-Smoczyk)
Let M be a Lagrangian submanifold of S3 × S3 Then we have
1 M is orientable
2 G (X ,Y ) is a normal vector (X ,Y tangential)
3 M is minimal
4 < h(X ,Y ), JZ > is totally symmetric
G is related to the canonical volume form ω by
ω(X ,Y ,Z ) =√
3g(JG (X ,Y ),Z ).
Almost complex surfaces
Basic properties of Lagrangian submanifolds
A 3-dimensional manifold M3 in S3 × S3 is called Lagrnagian if J mapsthe tangent space into the normal space.In view of the dimension this also implies that J of a normal vector istangent, i.e. J is an isomorphism between the tangent and the normalspace.
Theorem (Schafer-Smoczyk)
Let M be a Lagrangian submanifold of S3 × S3 Then we have
1 M is orientable
2 G (X ,Y ) is a normal vector (X ,Y tangential)
3 M is minimal
4 < h(X ,Y ), JZ > is totally symmetric
G is related to the canonical volume form ω by
ω(X ,Y ,Z ) =√
3g(JG (X ,Y ),Z ).
Almost complex surfaces
Basic properties of Lagrangian submanifolds
A 3-dimensional manifold M3 in S3 × S3 is called Lagrnagian if J mapsthe tangent space into the normal space.In view of the dimension this also implies that J of a normal vector istangent, i.e. J is an isomorphism between the tangent and the normalspace.
Theorem (Schafer-Smoczyk)
Let M be a Lagrangian submanifold of S3 × S3 Then we have
1 M is orientable
2 G (X ,Y ) is a normal vector (X ,Y tangential)
3 M is minimal
4 < h(X ,Y ), JZ > is totally symmetric
G is related to the canonical volume form ω by
ω(X ,Y ,Z ) =√
3g(JG (X ,Y ),Z ).
Almost complex surfaces
Basic properties of Lagrangian submanifolds
A 3-dimensional manifold M3 in S3 × S3 is called Lagrnagian if J mapsthe tangent space into the normal space.In view of the dimension this also implies that J of a normal vector istangent, i.e. J is an isomorphism between the tangent and the normalspace.
Theorem (Schafer-Smoczyk)
Let M be a Lagrangian submanifold of S3 × S3 Then we have
1 M is orientable
2 G (X ,Y ) is a normal vector (X ,Y tangential)
3 M is minimal
4 < h(X ,Y ), JZ > is totally symmetric
G is related to the canonical volume form ω by
ω(X ,Y ,Z ) =√
3g(JG (X ,Y ),Z ).
Almost complex surfaces
Basic properties of Lagrangian submanifolds
A 3-dimensional manifold M3 in S3 × S3 is called Lagrnagian if J mapsthe tangent space into the normal space.In view of the dimension this also implies that J of a normal vector istangent, i.e. J is an isomorphism between the tangent and the normalspace.
Theorem (Schafer-Smoczyk)
Let M be a Lagrangian submanifold of S3 × S3 Then we have
1 M is orientable
2 G (X ,Y ) is a normal vector (X ,Y tangential)
3 M is minimal
4 < h(X ,Y ), JZ > is totally symmetric
G is related to the canonical volume form ω by
ω(X ,Y ,Z ) =√
3g(JG (X ,Y ),Z ).
Almost complex surfaces
Basic properties of Lagrangian submanifolds
A 3-dimensional manifold M3 in S3 × S3 is called Lagrnagian if J mapsthe tangent space into the normal space.In view of the dimension this also implies that J of a normal vector istangent, i.e. J is an isomorphism between the tangent and the normalspace.
Theorem (Schafer-Smoczyk)
Let M be a Lagrangian submanifold of S3 × S3 Then we have
1 M is orientable
2 G (X ,Y ) is a normal vector (X ,Y tangential)
3 M is minimal
4 < h(X ,Y ), JZ > is totally symmetric
G is related to the canonical volume form ω by
ω(X ,Y ,Z ) =√
3g(JG (X ,Y ),Z ).
Almost complex surfaces
Basic properties of Lagrangian submanifolds
We can writePX = AX + JBX
We have that
1 A and B are symmetric operators
2 A2 + B2 = I
3 [A,B] = 0
These properties imply that A and B can be diagonalised simultaneouslyand that there exists a basis e1, e2, e3 of the tangent space at every pointsuch that
Pei = cos 2θiei + sin 2θiJei
Almost complex surfaces
Basic properties of Lagrangian submanifolds
We can writePX = AX + JBX
We have that
1 A and B are symmetric operators
2 A2 + B2 = I
3 [A,B] = 0
These properties imply that A and B can be diagonalised simultaneouslyand that there exists a basis e1, e2, e3 of the tangent space at every pointsuch that
Pei = cos 2θiei + sin 2θiJei
Almost complex surfaces
Basic properties of Lagrangian submanifolds
We can writePX = AX + JBX
We have that
1 A and B are symmetric operators
2 A2 + B2 = I
3 [A,B] = 0
These properties imply that A and B can be diagonalised simultaneouslyand that there exists a basis e1, e2, e3 of the tangent space at every pointsuch that
Pei = cos 2θiei + sin 2θiJei
Almost complex surfaces
Basic properties of Lagrangian submanifolds
We can writePX = AX + JBX
We have that
1 A and B are symmetric operators
2 A2 + B2 = I
3 [A,B] = 0
These properties imply that A and B can be diagonalised simultaneouslyand that there exists a basis e1, e2, e3 of the tangent space at every pointsuch that
Pei = cos 2θiei + sin 2θiJei
Almost complex surfaces
Basic properties of Lagrangian submanifolds
We can writePX = AX + JBX
We have that
1 A and B are symmetric operators
2 A2 + B2 = I
3 [A,B] = 0
These properties imply that A and B can be diagonalised simultaneouslyand that there exists a basis e1, e2, e3 of the tangent space at every pointsuch that
Pei = cos 2θiei + sin 2θiJei
Almost complex surfaces
Basic properties of Lagrangian submanifolds
Basic Equations
1 The tensor T (X ,Y ) = SJXY = −Jh(X ,Y ) is minimal andg(T (X ,Y ),Z ) is totally symmetric,
2 Gauss equation
R(X ,Y )Z =5
12
(g(Y ,Z )X − g(X ,Z )Y
)+
1
3
(g(AY ,Z )AX − g(AX ,Z )AY + g(BY ,Z )BX − g(BX ,Z )BY
)+ [SJX ,SJY ]Z .
3 Codazzi equation
∇h(X ,Y ,Z ))−∇h(Y ,X ,Z ) =
1
3
(g(AY ,Z )JBX − g(AX ,Z )JBY − g(BY ,Z )JAX + g(BX ,Z )JAY
)
Almost complex surfaces
Basic properties of Lagrangian submanifolds
Basic Equations
1 The tensor T (X ,Y ) = SJXY = −Jh(X ,Y ) is minimal andg(T (X ,Y ),Z ) is totally symmetric,
2 Gauss equation
R(X ,Y )Z =5
12
(g(Y ,Z )X − g(X ,Z )Y
)+
1
3
(g(AY ,Z )AX − g(AX ,Z )AY + g(BY ,Z )BX − g(BX ,Z )BY
)+ [SJX ,SJY ]Z .
3 Codazzi equation
∇h(X ,Y ,Z ))−∇h(Y ,X ,Z ) =
1
3
(g(AY ,Z )JBX − g(AX ,Z )JBY − g(BY ,Z )JAX + g(BX ,Z )JAY
)
Almost complex surfaces
Basic properties of Lagrangian submanifolds
Basic Equations
1 The tensor T (X ,Y ) = SJXY = −Jh(X ,Y ) is minimal andg(T (X ,Y ),Z ) is totally symmetric,
2 Gauss equation
R(X ,Y )Z =5
12
(g(Y ,Z )X − g(X ,Z )Y
)+
1
3
(g(AY ,Z )AX − g(AX ,Z )AY + g(BY ,Z )BX − g(BX ,Z )BY
)+ [SJX ,SJY ]Z .
3 Codazzi equation
∇h(X ,Y ,Z ))−∇h(Y ,X ,Z ) =
1
3
(g(AY ,Z )JBX − g(AX ,Z )JBY − g(BY ,Z )JAX + g(BX ,Z )JAY
)
Almost complex surfaces
Basic properties of Lagrangian submanifolds
Basic Equations
Covariant derivatives equations for A and B:
(∇XA)Y = BSJXY − Jh(X ,BY ) +1
2
(JG (X ,AY )− AJG (X ,Y )
),
(∇XB)Y = Jh(X ,AY )− ASJXY +1
2
(JG (X ,BY )− BJG (X ,Y )
).
Almost complex surfaces
Basic properties of Lagrangian submanifolds
Proposition
The sum of the angle functions vanishes modulo π
Almost complex surfaces
Basic properties of Lagrangian submanifolds
Lemma
The derivatives of the angles θi give the components of the secondfundamental form
Ei (θj) = −hijjexcept h3
12. The second fundamental form and covariant derivative arerelated by
hkij cos(θj − θk) =(√3
6εkij − ωk
ij
)sin(θj − θk).
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Elementary examples of Lagrangian submanifolds
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Schafer-Smoczyk
Example 1: f (g) = (g , 1).
df (E1(g))) = (gi , 0)(g ,1),
df (E2(g))) = (gj , 0)(g ,1),
df (E3(g))) = (−gk , 0)(g ,1).
Pdf (E1(g))) = (0, i)(g ,1),
Pdf (E2(g))) = (0, j)(g ,1),
Pdf (E3(g))) = (0,−k)(g ,1).
Jdf (E1(g))) = 1√3
(−gi ,−2i)(g ,1),
Jdf (E2(g))) = 1√3
(−gj ,−2j)(g ,1),
Jdf (E3(g))) = 1√3
(gk, 2k)(g ,1).
(2θ1, 2θ2, 2θ3) = ( 2π3 ,
2π3 ,
2π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Schafer-Smoczyk
Example 1: f (g) = (g , 1).
df (E1(g))) = (gi , 0)(g ,1),
df (E2(g))) = (gj , 0)(g ,1),
df (E3(g))) = (−gk , 0)(g ,1).
Pdf (E1(g))) = (0, i)(g ,1),
Pdf (E2(g))) = (0, j)(g ,1),
Pdf (E3(g))) = (0,−k)(g ,1).
Jdf (E1(g))) = 1√3
(−gi ,−2i)(g ,1),
Jdf (E2(g))) = 1√3
(−gj ,−2j)(g ,1),
Jdf (E3(g))) = 1√3
(gk, 2k)(g ,1).
(2θ1, 2θ2, 2θ3) = ( 2π3 ,
2π3 ,
2π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Schafer-Smoczyk
Example 1: f (g) = (g , 1).
df (E1(g))) = (gi , 0)(g ,1),
df (E2(g))) = (gj , 0)(g ,1),
df (E3(g))) = (−gk , 0)(g ,1).
Pdf (E1(g))) = (0, i)(g ,1),
Pdf (E2(g))) = (0, j)(g ,1),
Pdf (E3(g))) = (0,−k)(g ,1).
Jdf (E1(g))) = 1√3
(−gi ,−2i)(g ,1),
Jdf (E2(g))) = 1√3
(−gj ,−2j)(g ,1),
Jdf (E3(g))) = 1√3
(gk, 2k)(g ,1).
(2θ1, 2θ2, 2θ3) = ( 2π3 ,
2π3 ,
2π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Schafer-Smoczyk
Example 1: f (g) = (g , 1).
df (E1(g))) = (gi , 0)(g ,1),
df (E2(g))) = (gj , 0)(g ,1),
df (E3(g))) = (−gk , 0)(g ,1).
Pdf (E1(g))) = (0, i)(g ,1),
Pdf (E2(g))) = (0, j)(g ,1),
Pdf (E3(g))) = (0,−k)(g ,1).
Jdf (E1(g))) = 1√3
(−gi ,−2i)(g ,1),
Jdf (E2(g))) = 1√3
(−gj ,−2j)(g ,1),
Jdf (E3(g))) = 1√3
(gk, 2k)(g ,1).
(2θ1, 2θ2, 2θ3) = ( 2π3 ,
2π3 ,
2π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Schafer-Smoczyk
Example 1: f (g) = (g , 1).
df (E1(g))) = (gi , 0)(g ,1),
df (E2(g))) = (gj , 0)(g ,1),
df (E3(g))) = (−gk , 0)(g ,1).
Pdf (E1(g))) = (0, i)(g ,1),
Pdf (E2(g))) = (0, j)(g ,1),
Pdf (E3(g))) = (0,−k)(g ,1).
Jdf (E1(g))) = 1√3
(−gi ,−2i)(g ,1),
Jdf (E2(g))) = 1√3
(−gj ,−2j)(g ,1),
Jdf (E3(g))) = 1√3
(gk, 2k)(g ,1).
(2θ1, 2θ2, 2θ3) = ( 2π3 ,
2π3 ,
2π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Schafer-Smoczyk
Example 2: f (g) = (1, g)
df (E1(g)) = (0, gi)(1,g),
df (E2(g)) = (0, gj)(1,g),
df (E3(g)) = (0,−gk)(1,g).
Pdf (E1(g)) = (i , 0)(1,g),
Pdf (E2(g)) = (j , 0)(1,g),
Pdf (E3(g)) = (k, 0)(1,g).
Jdf (E1(g)) = 1√3
(2i , gi)(1,g),
Jdf (E2(g)) = 1√3
(2j , gj)(1,g)),
Jdf (E3(g)) = 1√3
(−2k, gk)(1,g).
(2θ1, 2θ2, 2θ3) = ( 4π3 ,
4π3 ,
4π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Schafer-Smoczyk
Example 2: f (g) = (1, g)
df (E1(g)) = (0, gi)(1,g),
df (E2(g)) = (0, gj)(1,g),
df (E3(g)) = (0,−gk)(1,g).
Pdf (E1(g)) = (i , 0)(1,g),
Pdf (E2(g)) = (j , 0)(1,g),
Pdf (E3(g)) = (k , 0)(1,g).
Jdf (E1(g)) = 1√3
(2i , gi)(1,g),
Jdf (E2(g)) = 1√3
(2j , gj)(1,g)),
Jdf (E3(g)) = 1√3
(−2k , gk)(1,g).
(2θ1, 2θ2, 2θ3) = ( 4π3 ,
4π3 ,
4π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Schafer-Smoczyk
Example 3: f (g) = (g , g)
df (E1(g)) = (gi , gi)(g ,g) = Pdf (E1(g))),
df (E2(g)) = (gj , gj)(g ,g) = Pdf (E2(g))),
df (E3(g)) = (−gk ,−gk)(g ,g) = Pdf (E2(g)).
Jdf (E1(g)) = 1√3
(gi ,−gi)(g ,g),
Jdf (E2(g)) = 1√3
(gj , gj)(g ,g)),
Jdf (E3(g)) = 1√3
(−gk , gk)(g ,g).
As P is the identity, we see that the angle functions vanish.
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Schafer-Smoczyk
Example 3: f (g) = (g , g)
df (E1(g)) = (gi , gi)(g ,g) = Pdf (E1(g))),
df (E2(g)) = (gj , gj)(g ,g) = Pdf (E2(g))),
df (E3(g)) = (−gk ,−gk)(g ,g) = Pdf (E2(g)).
Jdf (E1(g)) = 1√3
(gi ,−gi)(g ,g),
Jdf (E2(g)) = 1√3
(gj , gj)(g ,g)),
Jdf (E3(g)) = 1√3
(−gk , gk)(g ,g).
As P is the identity, we see that the angle functions vanish.
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 4: f (g) = (g , gi)
df (E1(g)) = (gi ,−g)(g ,gi),
df (E2(g)) = (gj ,−gk)(g ,gi),
df (E3(g)) = (−gk ,−gj)(g ,gi).
Pdf (E1(g)) = (gi ,−g)(g ,gi) = dF (E1(g)),
Pdf (E2(g)) = (gj ,−gk)(g ,gi) = −dF (E2(g)),
Pdf (E3(g)) = (−gk,−gj)(g ,gi) = −dF (E3(g)).
Jdf (E1(g)) = 1√3
(gi , g)(g ,gi),
Jdf (E2(g)) = −√
3(gj , gk)(g ,gi)),
Jdf (E3(g)) =√
3(gk ,−gj)(g ,gi).
(2θ1, 2θ2, 2θ3) = (0, π, π)
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 4: f (g) = (g , gi)
df (E1(g)) = (gi ,−g)(g ,gi),
df (E2(g)) = (gj ,−gk)(g ,gi),
df (E3(g)) = (−gk ,−gj)(g ,gi).
Pdf (E1(g)) = (gi ,−g)(g ,gi) = dF (E1(g)),
Pdf (E2(g)) = (gj ,−gk)(g ,gi) = −dF (E2(g)),
Pdf (E3(g)) = (−gk,−gj)(g ,gi) = −dF (E3(g)).
Jdf (E1(g)) = 1√3
(gi , g)(g ,gi),
Jdf (E2(g)) = −√
3(gj , gk)(g ,gi)),
Jdf (E3(g)) =√
3(gk ,−gj)(g ,gi).
(2θ1, 2θ2, 2θ3) = (0, π, π)
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 4: f (g) = (g , gi)
df (E1(g)) = (gi ,−g)(g ,gi),
df (E2(g)) = (gj ,−gk)(g ,gi),
df (E3(g)) = (−gk ,−gj)(g ,gi).
Pdf (E1(g)) = (gi ,−g)(g ,gi) = dF (E1(g)),
Pdf (E2(g)) = (gj ,−gk)(g ,gi) = −dF (E2(g)),
Pdf (E3(g)) = (−gk,−gj)(g ,gi) = −dF (E3(g)).
Jdf (E1(g)) = 1√3
(gi , g)(g ,gi),
Jdf (E2(g)) = −√
3(gj , gk)(g ,gi)),
Jdf (E3(g)) =√
3(gk ,−gj)(g ,gi).
(2θ1, 2θ2, 2θ3) = (0, π, π)
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 4: f (g) = (g , gi)
df (E1(g)) = (gi ,−g)(g ,gi),
df (E2(g)) = (gj ,−gk)(g ,gi),
df (E3(g)) = (−gk ,−gj)(g ,gi).
Pdf (E1(g)) = (gi ,−g)(g ,gi) = dF (E1(g)),
Pdf (E2(g)) = (gj ,−gk)(g ,gi) = −dF (E2(g)),
Pdf (E3(g)) = (−gk,−gj)(g ,gi) = −dF (E3(g)).
Jdf (E1(g)) = 1√3
(gi , g)(g ,gi),
Jdf (E2(g)) = −√
3(gj , gk)(g ,gi)),
Jdf (E3(g)) =√
3(gk ,−gj)(g ,gi).
(2θ1, 2θ2, 2θ3) = (0, π, π)
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 4: f (g) = (g , gi)
df (E1(g)) = (gi ,−g)(g ,gi),
df (E2(g)) = (gj ,−gk)(g ,gi),
df (E3(g)) = (−gk ,−gj)(g ,gi).
Pdf (E1(g)) = (gi ,−g)(g ,gi) = dF (E1(g)),
Pdf (E2(g)) = (gj ,−gk)(g ,gi) = −dF (E2(g)),
Pdf (E3(g)) = (−gk,−gj)(g ,gi) = −dF (E3(g)).
Jdf (E1(g)) = 1√3
(gi , g)(g ,gi),
Jdf (E2(g)) = −√
3(gj , gk)(g ,gi)),
Jdf (E3(g)) =√
3(gk ,−gj)(g ,gi).
(2θ1, 2θ2, 2θ3) = (0, π, π)
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 5: f (g) = (g−1, gig−1)
df (E1(g))) = (−ig−1, 0)(g−1,gig−1),
df (E2(g))) = (−jg−1,−2gkg−1)(g−1,gig−1),
df (E3(g))) = (kg−1,−2gjg−1)(g−1,gig−1).
Pdf (E1(g))) = (0, 1)(g−1,gig−1),
Pdf (E2(g))) = (−2jg−1,−gkg−1)(g−1,gig−1),
Pdf (E3(g))) = (2kg−1,−gjg−1)(g−1,gig−1).
Jdf (E1(g))) = 1√3
(ig−1,−2)(g−1,gig−1),
Jdf (E2(g))) = 1√3
(−jg−1, 0)(g−1,gig−1),
Jdf (E3(g))) = 1√3
(3kg−1, 0)(g−1,gig−1).
(2θ1, 2θ2, 2θ3) = (π3 ,π3 ,
4π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 5: f (g) = (g−1, gig−1)
df (E1(g))) = (−ig−1, 0)(g−1,gig−1),
df (E2(g))) = (−jg−1,−2gkg−1)(g−1,gig−1),
df (E3(g))) = (kg−1,−2gjg−1)(g−1,gig−1).
Pdf (E1(g))) = (0, 1)(g−1,gig−1),
Pdf (E2(g))) = (−2jg−1,−gkg−1)(g−1,gig−1),
Pdf (E3(g))) = (2kg−1,−gjg−1)(g−1,gig−1).
Jdf (E1(g))) = 1√3
(ig−1,−2)(g−1,gig−1),
Jdf (E2(g))) = 1√3
(−jg−1, 0)(g−1,gig−1),
Jdf (E3(g))) = 1√3
(3kg−1, 0)(g−1,gig−1).
(2θ1, 2θ2, 2θ3) = (π3 ,π3 ,
4π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 5: f (g) = (g−1, gig−1)
df (E1(g))) = (−ig−1, 0)(g−1,gig−1),
df (E2(g))) = (−jg−1,−2gkg−1)(g−1,gig−1),
df (E3(g))) = (kg−1,−2gjg−1)(g−1,gig−1).
Pdf (E1(g))) = (0, 1)(g−1,gig−1),
Pdf (E2(g))) = (−2jg−1,−gkg−1)(g−1,gig−1),
Pdf (E3(g))) = (2kg−1,−gjg−1)(g−1,gig−1).
Jdf (E1(g))) = 1√3
(ig−1,−2)(g−1,gig−1),
Jdf (E2(g))) = 1√3
(−jg−1, 0)(g−1,gig−1),
Jdf (E3(g))) = 1√3
(3kg−1, 0)(g−1,gig−1).
(2θ1, 2θ2, 2θ3) = (π3 ,π3 ,
4π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 6: f (g) = (gig−1, g−1)
So this is the previous immersions withboth components interchanged. Then
df (E1(g))) = (0,−ig−1)(gig−1,g−1),
df (E2(g))) = (−2gkg−1,−jg−1)(gig−1,g−1),
df (E3(g))) = (−2gjg−1, kg−1)(gig−1,g−1).
Pdf (E1(g))) = (1, 0)(gig−1,g−1),
Pdf (E2(g))) = (−gkg−1,−2jg−1)(gig−1,g−1),
Pdf (E3(g))) = (−gjg−1, 2kg−1)(gig−1,g−1).
Jdf (E1(g))) = 1√3
(2, ig−1)(gig−1,g−1),
Jdf (E2(g))) = 1√3
(0, 3jg−1)(gig−1,g−1),
Jdf (E3(g))) = 1√3
(0,−3kg−1)(gig−1,g−1).
(2θ1, 2θ2, 2θ3) = ( 2π3 ,
5π3 ,
5π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 6: f (g) = (gig−1, g−1) So this is the previous immersions withboth components interchanged. Then
df (E1(g))) = (0,−ig−1)(gig−1,g−1),
df (E2(g))) = (−2gkg−1,−jg−1)(gig−1,g−1),
df (E3(g))) = (−2gjg−1, kg−1)(gig−1,g−1).
Pdf (E1(g))) = (1, 0)(gig−1,g−1),
Pdf (E2(g))) = (−gkg−1,−2jg−1)(gig−1,g−1),
Pdf (E3(g))) = (−gjg−1, 2kg−1)(gig−1,g−1).
Jdf (E1(g))) = 1√3
(2, ig−1)(gig−1,g−1),
Jdf (E2(g))) = 1√3
(0, 3jg−1)(gig−1,g−1),
Jdf (E3(g))) = 1√3
(0,−3kg−1)(gig−1,g−1).
(2θ1, 2θ2, 2θ3) = ( 2π3 ,
5π3 ,
5π3 )
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 7: f (g) = (gig−1, gjg−1)
For the tangent map wehave df (E1) = (0, 2gkg−1), df (E2) = (−2gkg−1, 0),df (E3) = 2(−gjg−1, gig−1).Also Jdf (E1) = 2√
3(2, gkg−1), Jdf (E2) = 2√
3(gkg−1,−2) and
Jdf (E3) = − 2√3
(gjg−1, gig−1).
We also have
Pdf (E1) = (2, 0) = − 12 (df (E1)−
√3Jdf (E2)),
Pdf (E2) = (0, 2) = − 12 (df (E1) +
√3Jdf (E2)),
Pdf (E3) = −2(gjg−1,−gig−1) = df (E3).
The angles 2θi are thus equal to 0, 2π3 and 4π
3 .
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 7: f (g) = (gig−1, gjg−1) For the tangent map wehave df (E1) = (0, 2gkg−1), df (E2) = (−2gkg−1, 0),df (E3) = 2(−gjg−1, gig−1).
Also Jdf (E1) = 2√3
(2, gkg−1), Jdf (E2) = 2√3
(gkg−1,−2) and
Jdf (E3) = − 2√3
(gjg−1, gig−1).
We also have
Pdf (E1) = (2, 0) = − 12 (df (E1)−
√3Jdf (E2)),
Pdf (E2) = (0, 2) = − 12 (df (E1) +
√3Jdf (E2)),
Pdf (E3) = −2(gjg−1,−gig−1) = df (E3).
The angles 2θi are thus equal to 0, 2π3 and 4π
3 .
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 7: f (g) = (gig−1, gjg−1) For the tangent map wehave df (E1) = (0, 2gkg−1), df (E2) = (−2gkg−1, 0),df (E3) = 2(−gjg−1, gig−1).Also Jdf (E1) = 2√
3(2, gkg−1), Jdf (E2) = 2√
3(gkg−1,−2) and
Jdf (E3) = − 2√3
(gjg−1, gig−1).
We also have
Pdf (E1) = (2, 0) = − 12 (df (E1)−
√3Jdf (E2)),
Pdf (E2) = (0, 2) = − 12 (df (E1) +
√3Jdf (E2)),
Pdf (E3) = −2(gjg−1,−gig−1) = df (E3).
The angles 2θi are thus equal to 0, 2π3 and 4π
3 .
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 7: f (g) = (gig−1, gjg−1) For the tangent map wehave df (E1) = (0, 2gkg−1), df (E2) = (−2gkg−1, 0),df (E3) = 2(−gjg−1, gig−1).Also Jdf (E1) = 2√
3(2, gkg−1), Jdf (E2) = 2√
3(gkg−1,−2) and
Jdf (E3) = − 2√3
(gjg−1, gig−1).
We also have
Pdf (E1) = (2, 0) = − 12 (df (E1)−
√3Jdf (E2)),
Pdf (E2) = (0, 2) = − 12 (df (E1) +
√3Jdf (E2)),
Pdf (E3) = −2(gjg−1,−gig−1) = df (E3).
The angles 2θi are thus equal to 0, 2π3 and 4π
3 .
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
Moroianu-Semmelmann
Example 7: f (g) = (gig−1, gjg−1) For the tangent map wehave df (E1) = (0, 2gkg−1), df (E2) = (−2gkg−1, 0),df (E3) = 2(−gjg−1, gig−1).Also Jdf (E1) = 2√
3(2, gkg−1), Jdf (E2) = 2√
3(gkg−1,−2) and
Jdf (E3) = − 2√3
(gjg−1, gig−1).
We also have
Pdf (E1) = (2, 0) = − 12 (df (E1)−
√3Jdf (E2)),
Pdf (E2) = (0, 2) = − 12 (df (E1) +
√3Jdf (E2)),
Pdf (E3) = −2(gjg−1,−gig−1) = df (E3).
The angles 2θi are thus equal to 0, 2π3 and 4π
3 .
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
New
Example 8. Consider theimmersion f : R3 → S3 × S3 : (u, v , w) 7→ (p(u, w), q(u, v)) where p andq are constant mean curvature torii in S3 given by
p = (cos(u) cos(w), cos(u) sin(w), sin(u) cos(w), ε1 sin(u) sin(w)),
q = (cos(u) cos(v), cos(u) sin(v), sin(u) cos(v), ε2 sin(u) sin(w))d ,
where d is the unit quaternion given by
( 1√2,− 1√
2, 0, 0)
Straightforward computations give that u, v ,w determined by u =√
32 u,
v =√
32 v and w =
√3
2 w are the standard flat coordinates. So it is animmersion of a torus. The angles 2θi are again equal to 0, 2π
3 and 4π3 .
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
New
Example 8. Consider theimmersion f : R3 → S3 × S3 : (u, v , w) 7→ (p(u, w), q(u, v)) where p andq are constant mean curvature torii in S3 given by
p = (cos(u) cos(w), cos(u) sin(w), sin(u) cos(w), ε1 sin(u) sin(w)),
q = (cos(u) cos(v), cos(u) sin(v), sin(u) cos(v), ε2 sin(u) sin(w))d ,
where d is the unit quaternion given by
( 1√2,− 1√
2, 0, 0)
Straightforward computations give that u, v ,w determined by u =√
32 u,
v =√
32 v and w =
√3
2 w are the standard flat coordinates. So it is animmersion of a torus. The angles 2θi are again equal to 0, 2π
3 and 4π3 .
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
New
Example 8. Consider theimmersion f : R3 → S3 × S3 : (u, v , w) 7→ (p(u, w), q(u, v)) where p andq are constant mean curvature torii in S3 given by
p = (cos(u) cos(w), cos(u) sin(w), sin(u) cos(w), ε1 sin(u) sin(w)),
q = (cos(u) cos(v), cos(u) sin(v), sin(u) cos(v), ε2 sin(u) sin(w))d ,
where d is the unit quaternion given by
( 1√2,− 1√
2, 0, 0)
Straightforward computations give that u, v ,w determined by u =√
32 u,
v =√
32 v and w =
√3
2 w are the standard flat coordinates. So it is animmersion of a torus. The angles 2θi are again equal to 0, 2π
3 and 4π3 .
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
New
Example 8. Consider theimmersion f : R3 → S3 × S3 : (u, v , w) 7→ (p(u, w), q(u, v)) where p andq are constant mean curvature torii in S3 given by
p = (cos(u) cos(w), cos(u) sin(w), sin(u) cos(w), ε1 sin(u) sin(w)),
q = (cos(u) cos(v), cos(u) sin(v), sin(u) cos(v), ε2 sin(u) sin(w))d ,
where d is the unit quaternion given by
( 1√2,− 1√
2, 0, 0)
Straightforward computations give that u, v ,w determined by u =√
32 u,
v =√
32 v and w =
√3
2 w are the standard flat coordinates. So it is animmersion of a torus.
The angles 2θi are again equal to 0, 2π3 and 4π
3 .
Almost complex surfaces
Elementary examples of Lagrangian submanifolds
New
Example 8. Consider theimmersion f : R3 → S3 × S3 : (u, v , w) 7→ (p(u, w), q(u, v)) where p andq are constant mean curvature torii in S3 given by
p = (cos(u) cos(w), cos(u) sin(w), sin(u) cos(w), ε1 sin(u) sin(w)),
q = (cos(u) cos(v), cos(u) sin(v), sin(u) cos(v), ε2 sin(u) sin(w))d ,
where d is the unit quaternion given by
( 1√2,− 1√
2, 0, 0)
Straightforward computations give that u, v ,w determined by u =√
32 u,
v =√
32 v and w =
√3
2 w are the standard flat coordinates. So it is animmersion of a torus. The angles 2θi are again equal to 0, 2π
3 and 4π3 .
Almost complex surfaces
Totally geodesic Lagrangian submanifolds
Totally geodesic Lagrangian submanifolds
Theorem
Let M3 be a totally geodesic Lagrangian immersion in S3 × S3 then M islocally congruent with either
1 g 7→ (g , 1)
2 g 7→ (1, g)
3 g 7→ (g , g)
4 g 7→ (g , gi)
5 g 7→ (g−1, gig−1)
6 g 7→ (gig−1, g−1)
Almost complex surfaces
Lagrangian submanifolds with constant sectional curvature
Lagrangian submanifolds with constant sectionalcurvature
Theorem
Let M3 be a totally geodesic Lagrangian immersion in S3 × S3 then M istotally geodesic or locally congruent with either
1 g 7→ (gig−1, gjg−1)
2 the flat torus described earlier.
