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Motivation: the title term by term Results

The algebra of the parallel endomorphisms of a germ of pseudo-Riemannian metric

Charles Boubel Universit´ e de Strasbourg August 27, 2012 PADGE, KU Leuven

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Motivation: the title term by term Results

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Motivation: the title term by term Results

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Motivation: the title term by term

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Pseudo-Riemannian metric A nondegenerate bilinear form on some manifold M, with sign(g) = (r, s), r = 0, s = 0. Endomorphism, more precisely endomorphism field A section U of End(TM). So at each point p ∈ M, U|p ∈ End(TpM). Parallel

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Pseudo-Riemannian metric A nondegenerate bilinear form on some manifold M, with sign(g) = (r, s), r = 0, s = 0. Endomorphism, more precisely endomorphism field A section U of End(TM). So at each point p ∈ M, U|p ∈ End(TpM). Parallel

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Pseudo-Riemannian metric A nondegenerate bilinear form on some manifold M, with sign(g) = (r, s), r = 0, s = 0. Endomorphism, more precisely endomorphism field A section U of End(TM). So at each point p ∈ M, U|p ∈ End(TpM). Parallel

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Algebra Let us set A := {U ∈ Γ(End(TM)); DU = 0}, equivalently A := {U ∈ End(TpM); ∀h ∈ Hp, h ◦ U = U ◦ h}. This is an associative algebra: Id ∈ A, A is stable by sum and composition. Germs In other terms, the work presented here is local. Think that M is a small ball around some point p.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Algebra Let us set A := {U ∈ Γ(End(TM)); DU = 0}, equivalently A := {U ∈ End(TpM); ∀h ∈ Hp, h ◦ U = U ◦ h}. This is an associative algebra: Id ∈ A, A is stable by sum and composition. Germs In other terms, the work presented here is local. Think that M is a small ball around some point p.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Back to ”Pseudo-Riemannian”: why ”pseudo” ? What if g is Riemannian i.e. positive definite ? The story is very simple : – A is a field, more precisely A = R. Id (generic case), A = J ≃ C (K¨ ahler metrics) or A = J1, J2, J3 ≃ H (hyperk¨ ahler metrics). – because of a theorem of de Rham: If a subspace Ep of TpM is Hp-stable, so is E ⊥

p , and those give

rise to integrable distributions E ⊕ E ⊥, hence to integral foliations E and E⊥.

• Theorem. M is a Riemannian product:

(M, g) ≃ (Ep, g1) × (E⊥

p , g2)

corresponding to a decomposition H = H1 × H2 of H.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Back to ”Pseudo-Riemannian”: why ”pseudo” ? What if g is Riemannian i.e. positive definite ? The story is very simple : – A is a field, more precisely A = R. Id (generic case), A = J ≃ C (K¨ ahler metrics) or A = J1, J2, J3 ≃ H (hyperk¨ ahler metrics). – because of a theorem of de Rham: If a subspace Ep of TpM is Hp-stable, so is E ⊥

p , and those give

rise to integrable distributions E ⊕ E ⊥, hence to integral foliations E and E⊥.

• Theorem. M is a Riemannian product:

(M, g) ≃ (Ep, g1) × (E⊥

p , g2)

corresponding to a decomposition H = H1 × H2 of H.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Back to ”Pseudo-Riemannian”: why ”pseudo” ? What if g is Riemannian i.e. positive definite ? The story is very simple : – A is a field, more precisely A = R. Id (generic case), A = J ≃ C (K¨ ahler metrics) or A = J1, J2, J3 ≃ H (hyperk¨ ahler metrics). – because of a theorem of de Rham: If a subspace Ep of TpM is Hp-stable, so is E ⊥

p , and those give

rise to integrable distributions E ⊕ E ⊥, hence to integral foliations E and E⊥.

• Theorem. M is a Riemannian product:

(M, g) ≃ (Ep, g1) × (E⊥

p , g2)

corresponding to a decomposition H = H1 × H2 of H.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Back to ”Pseudo-Riemannian”: why ”pseudo” ? What if g is Riemannian i.e. positive definite ? The story is very simple : – A is a field, more precisely A = R. Id (generic case), A = J ≃ C (K¨ ahler metrics) or A = J1, J2, J3 ≃ H (hyperk¨ ahler metrics). – because of a theorem of de Rham: If a subspace Ep of TpM is Hp-stable, so is E ⊥

p , and those give

rise to integrable distributions E ⊕ E ⊥, hence to integral foliations E and E⊥.

• Theorem. M is a Riemannian product:

(M, g) ≃ (Ep, g1) × (E⊥

p , g2)

corresponding to a decomposition H = H1 × H2 of H.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Back to ”Pseudo-Riemannian”: why ”pseudo” ? What if g is Riemannian i.e. positive definite ? The story is very simple : – A is a field, more precisely A = R. Id (generic case), A = J ≃ C (K¨ ahler metrics) or A = J1, J2, J3 ≃ H (hyperk¨ ahler metrics). – because of a theorem of de Rham: If a subspace Ep of TpM is Hp-stable, so is E ⊥

p , and those give

rise to integrable distributions E ⊕ E ⊥, hence to integral foliations E and E⊥.

• Theorem. M is a Riemannian product:

(M, g) ≃ (Ep, g1) × (E⊥

p , g2)

corresponding to a decomposition H = H1 × H2 of H.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Consequences of this theorem of de Rham, if g is Riemannian – You may suppose that Hp acts irreducibly on TpM: else (M, g) is a Riemannian product, and you may consider each factor independently. – With this assumption, elementary linear algebra arguments show A ≃ R, A ≃ C or A ≃ H as announced (e.g. think that no nilpotent endomorphism N may be parallel, as ker N would be H-stable. Similarly, not more than one real eigenvalue is possible.)

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Consequences of this theorem of de Rham, if g is Riemannian – You may suppose that Hp acts irreducibly on TpM: else (M, g) is a Riemannian product, and you may consider each factor independently. – With this assumption, elementary linear algebra arguments show A ≃ R, A ≃ C or A ≃ H as announced (e.g. think that no nilpotent endomorphism N may be parallel, as ker N would be H-stable. Similarly, not more than one real eigenvalue is possible.)

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Consequences of this theorem of de Rham, if g is Riemannian – You may suppose that Hp acts irreducibly on TpM: else (M, g) is a Riemannian product, and you may consider each factor independently. – With this assumption, elementary linear algebra arguments show A ≃ R, A ≃ C or A ≃ H as announced (e.g. think that no nilpotent endomorphism N may be parallel, as ker N would be H-stable. Similarly, not more than one real eigenvalue is possible.)

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

What does fail in the pseudo-Riemannian case ? If a subspace Ep of TpM is Hp-stable, so is E ⊥

p , and those give

rise to integrable distributions E ⊕ E ⊥, hence to integral foliations E and E⊥.

• Theorem. M is a Riemannian product:

(M, g) ≃ (Ep, g1) × (E⊥

p , g2)

corresponding to a decomposition H = H1 × H2 of H. So You may suppose that H acts irreducibly: else (M, g) is a Riemannian product, and you may consider each factor independently.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

What does fail in the pseudo-Riemannian case ? If a subspace Ep of TpM is Hp-stable, so is E ⊥

p , and those give

rise to integrable distributions E ⊕ E ⊥, hence to integral foliations E and E⊥.

• Theorem. M is a Riemannian product:

(M, g) ≃ (Ep, g1) × (E⊥

p , g2)

corresponding to a decomposition H = H1 × H2 of H. So You may suppose that H acts irreducibly: else (M, g) is a Riemannian product, and you may consider each factor independently.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

What does fail in the pseudo-Riemannian case ? If a subspace Ep of TpM is Hp-stable, so is E ⊥

p , and those give

rise to integrable distributions E ⊕ E ⊥, hence to integral foliations E and E⊥.

• Theorem. M is a Riemannian product:

(M, g) ≃ (Ep, g1) × (E⊥

p , g2)

corresponding to a decomposition H = H1 × H2 of H. So You may suppose that H acts irreducibly: else (M, g) is a Riemannian product, and you may consider each factor independently.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Possibly, E ∩ E ⊥ {0}. So the action of H may not be supposed to be irreducible in general, but only indecomposable (in an orthogonal sum of stable subspaces). In particular, a miscellany of parallel endomorphisms may appear, for some well-chosen pseudo-Riemannian metrics g.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Sum up of the motivations

◮ Each possible new form of A, with respect to the Riemannian

case, corresponds to a new local geometric structure, a new family of holonomy groups. Which are they ? To which sets

• f germs of metrics do they correspond ?

◮ This is a means to investigate pseudo-Riemannian metrics

with a non-irreducible holonomy representation, which are very little known.

◮ Is the arrow g → D injective ? Two metrics g and g′ share

the same Levi Civita connection iff g′ = g( · , U · ) with U parallel and self adjoint.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Sum up of the motivations

◮ Each possible new form of A, with respect to the Riemannian

case, corresponds to a new local geometric structure, a new family of holonomy groups. Which are they ? To which sets

• f germs of metrics do they correspond ?

◮ This is a means to investigate pseudo-Riemannian metrics

with a non-irreducible holonomy representation, which are very little known.

◮ Is the arrow g → D injective ? Two metrics g and g′ share

the same Levi Civita connection iff g′ = g( · , U · ) with U parallel and self adjoint.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Sum up of the motivations

◮ Each possible new form of A, with respect to the Riemannian

case, corresponds to a new local geometric structure, a new family of holonomy groups. Which are they ? To which sets

• f germs of metrics do they correspond ?

◮ This is a means to investigate pseudo-Riemannian metrics

with a non-irreducible holonomy representation, which are very little known.

◮ Is the arrow g → D injective ? Two metrics g and g′ share

the same Levi Civita connection iff g′ = g( · , U · ) with U parallel and self adjoint.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Sum up of the motivations

◮ Each possible new form of A, with respect to the Riemannian

case, corresponds to a new local geometric structure, a new family of holonomy groups. Which are they ? To which sets

• f germs of metrics do they correspond ?

◮ This is a means to investigate pseudo-Riemannian metrics

with a non-irreducible holonomy representation, which are very little known.

◮ Is the arrow g → D injective ? Two metrics g and g′ share

the same Levi Civita connection iff g′ = g( · , U · ) with U parallel and self adjoint.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Results

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

The decomposition of A into its semi-simple part and its radical As any associative algebra: A = As ⋉ Rad(A) (Wedderburn – Malˇ cev). Rad(A) is an ideal consisting of nilpotent, simultaneously upper triangular elements. Of course, necessarily Rad(A) = {0} in the Riemannian case.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

The semi-simple algebra As may be of eight different types Among them, As ≃ R ⊕ R, As ≃ C ⊕ C, As ≃ M2(R), As ≃ M2(C) etc. They are obtained by a mixture of parallel complex and paracomplex structures. I do not give any more details here.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Parallel nilpotent endomorphisms Algebras of simultaneously upper triangular nilpotent matrices are very complicated objects, any classification is out of reach. Here, the first natural question is: if N is some integrable field of nilpotent endomorphisms, is GN := {metrics g; DN = 0 and N∗ = N} non empty ? If yes, how to parametrise it ? Remark 1. Integrable = ”constant, in well-chosen coordinates”. Ex.: a complex structure, as opposed to an almost complex one. Remark 2. The case N∗ = −N is more complicated and not exposed here.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Parallel nilpotent endomorphisms Algebras of simultaneously upper triangular nilpotent matrices are very complicated objects, any classification is out of reach. Here, the first natural question is: if N is some integrable field of nilpotent endomorphisms, is GN := {metrics g; DN = 0 and N∗ = N} non empty ? If yes, how to parametrise it ? Remark 1. Integrable = ”constant, in well-chosen coordinates”. Ex.: a complex structure, as opposed to an almost complex one. Remark 2. The case N∗ = −N is more complicated and not exposed here.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Parallel nilpotent endomorphisms Algebras of simultaneously upper triangular nilpotent matrices are very complicated objects, any classification is out of reach. Here, the first natural question is: if N is some integrable field of nilpotent endomorphisms, is GN := {metrics g; DN = 0 and N∗ = N} non empty ? If yes, how to parametrise it ? Remark 1. Integrable = ”constant, in well-chosen coordinates”. Ex.: a complex structure, as opposed to an almost complex one. Remark 2. The case N∗ = −N is more complicated and not exposed here.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Parallel nilpotent endomorphisms Algebras of simultaneously upper triangular nilpotent matrices are very complicated objects, any classification is out of reach. Here, the first natural question is: if N is some integrable field of nilpotent endomorphisms, is GN := {metrics g; DN = 0 and N∗ = N} non empty ? If yes, how to parametrise it ? Remark 1. Integrable = ”constant, in well-chosen coordinates”. Ex.: a complex structure, as opposed to an almost complex one. Remark 2. The case N∗ = −N is more complicated and not exposed here.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Reminder: the K¨ ahler case If J is a complex structure on M, the local set of K¨ ahler metrics: GJ := {metrics g; DJ = 0 and J∗ = −J} is given by: g ∈ GJ if and only if, in complex coordinates (zj)j: g ∂ ∂zj , ∂ ∂zk

• =

∂2u ∂zj∂zk , where u is is a real function (the K¨ ahler potential).

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Let us change subject If J is a complex structure on M, and f a function: f : (M, J) → C = R[i] = R[X]/(X 2 + 1), f is said to be holomorphic if: df ◦ J = i df . Let us adapt this notion If N is an integrable field of nilpotent endomorphisms on M, and f a function: f : (M, J) → R[ν] = R[X]/(X n), we will say that f is nilomorphic if: df ◦ N = ν df .

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Let us change subject If J is a complex structure on M, and f a function: f : (M, J) → C = R[i] = R[X]/(X 2 + 1), f is said to be holomorphic if: df ◦ J = i df . Let us adapt this notion If N is an integrable field of nilpotent endomorphisms on M, and f a function: f : (M, J) → R[ν] = R[X]/(X n), we will say that f is nilomorphic if: df ◦ N = ν df .

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Let us change subject If J is a complex structure on M, and f a function: f : (M, J) → C = R[i] = R[X]/(X 2 + 1), f is said to be holomorphic if: df ◦ J = i df . Let us adapt this notion If N is an integrable field of nilpotent endomorphisms on M, and f a function: f : (M, J) → R[ν] = R[X]/(X n), we will say that f is nilomorphic if: df ◦ N = ν df .

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Let us change subject If J is a complex structure on M, and f a function: f : (M, J) → C = R[i] = R[X]/(X 2 + 1), f is said to be holomorphic if: df ◦ J = i df . Let us adapt this notion If N is an integrable field of nilpotent endomorphisms on M, and f a function: f : (M, J) → R[ν] = R[X]/(X n), we will say that f is nilomorphic if: df ◦ N = ν df .

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Let us change subject If J is a complex structure on M, and f a function: f : (M, J) → C = R[i] = R[X]/(X 2 + 1), f is said to be holomorphic if: df ◦ J = i df . Let us adapt this notion If N is an integrable field of nilpotent endomorphisms on M, and f a function: f : (M, J) → R[ν] = R[X]/(X n), we will say that f is nilomorphic if: df ◦ N = ν df .

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Of course, R[ν] is not a field, it is a ring, with zero divisers. But some parts of the theory of holomorphic functions have counterparts in this new framework, and will be useful. Similarly, all tensors, e.g. bilinear forms, have R[ν]-valued, nilomorphic versions. So we may speak of ”nilomorphic metrics”. I do not give details here. The main results are: Theorem A about nilomorphic functions Theorem B linking GN with nilomorphic metrics Corollary of Theorems A and B, parametrising GN Theorem C involving As to enhance Theorem B

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Of course, R[ν] is not a field, it is a ring, with zero divisers. But some parts of the theory of holomorphic functions have counterparts in this new framework, and will be useful. Similarly, all tensors, e.g. bilinear forms, have R[ν]-valued, nilomorphic versions. So we may speak of ”nilomorphic metrics”. I do not give details here. The main results are: Theorem A about nilomorphic functions Theorem B linking GN with nilomorphic metrics Corollary of Theorems A and B, parametrising GN Theorem C involving As to enhance Theorem B

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Of course, R[ν] is not a field, it is a ring, with zero divisers. But some parts of the theory of holomorphic functions have counterparts in this new framework, and will be useful. Similarly, all tensors, e.g. bilinear forms, have R[ν]-valued, nilomorphic versions. So we may speak of ”nilomorphic metrics”. I do not give details here. The main results are: Theorem A about nilomorphic functions Theorem B linking GN with nilomorphic metrics Corollary of Theorems A and B, parametrising GN Theorem C involving As to enhance Theorem B

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Of course, R[ν] is not a field, it is a ring, with zero divisers. But some parts of the theory of holomorphic functions have counterparts in this new framework, and will be useful. Similarly, all tensors, e.g. bilinear forms, have R[ν]-valued, nilomorphic versions. So we may speak of ”nilomorphic metrics”. I do not give details here. The main results are: Theorem A about nilomorphic functions Theorem B linking GN with nilomorphic metrics Corollary of Theorems A and B, parametrising GN Theorem C involving As to enhance Theorem B

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Of course, R[ν] is not a field, it is a ring, with zero divisers. But some parts of the theory of holomorphic functions have counterparts in this new framework, and will be useful. Similarly, all tensors, e.g. bilinear forms, have R[ν]-valued, nilomorphic versions. So we may speak of ”nilomorphic metrics”. I do not give details here. The main results are: Theorem A about nilomorphic functions Theorem B linking GN with nilomorphic metrics Corollary of Theorems A and B, parametrising GN Theorem C involving As to enhance Theorem B

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Theorem A

◮ We introduce the integral foliation I of Im N, and coordinates

adapted to N: (xi, (yi,a)n−1

a=1)i, ◮ and ”nilomorphic coordinates”:

zi = xi + νyi,1 + ν2yi,2 + . . . + νn−1yi,n−1 = xi + (νyi).

◮ Theorem A. A function f : (M, N) → R[ν] is nilomorphic if

and only if, in local nilomorphic coordinates: f =

• α

1 α! ∂|α|ˇ f ∂xα (νy)α, where ˇ f is the value of f along the transversal {(νy) = 0} to the foliation I. This ˇ f is a function of the xi (and must satisfy some conditions).

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Theorem A

◮ We introduce the integral foliation I of Im N, and coordinates

adapted to N: (xi, (yi,a)n−1

a=1)i, ◮ and ”nilomorphic coordinates”:

zi = xi + νyi,1 + ν2yi,2 + . . . + νn−1yi,n−1 = xi + (νyi).

◮ Theorem A. A function f : (M, N) → R[ν] is nilomorphic if

and only if, in local nilomorphic coordinates: f =

• α

1 α! ∂|α|ˇ f ∂xα (νy)α, where ˇ f is the value of f along the transversal {(νy) = 0} to the foliation I. This ˇ f is a function of the xi (and must satisfy some conditions).

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Theorem A

◮ We introduce the integral foliation I of Im N, and coordinates

adapted to N: (xi, (yi,a)n−1

a=1)i, ◮ and ”nilomorphic coordinates”:

zi = xi + νyi,1 + ν2yi,2 + . . . + νn−1yi,n−1 = xi + (νyi).

◮ Theorem A. A function f : (M, N) → R[ν] is nilomorphic if

and only if, in local nilomorphic coordinates: f =

• α

1 α! ∂|α|ˇ f ∂xα (νy)α, where ˇ f is the value of f along the transversal {(νy) = 0} to the foliation I. This ˇ f is a function of the xi (and must satisfy some conditions).

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Remark: similarity with the expansion of holomorphic functions Once you know all the derivatives of a holomorphic function f at some point p, you know it everywhere by its expansion as a power series: f =

• α

1 α! ∂|α|f ∂zα (p)zα. Notice that: – {p} is a transversal to the integral leaf of Im J = TM, – the coordinates zi parametrise this integral leaf. With this point of view, the expansion theorem is the same for nilomorphic functions. The obtained function is smooth in the xi, and polynomial in the yi,a.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

(A remark) There is a link with functions between spaces of jets. See the very recent work of W. Bertram and A. Souvay in Nancy (Arxiv, November 2011)

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Another parallel with complex Calculus On a K¨ ahler manifold (M, J, g), one introduces: h := g( · , · ) + ig( · , J · ) with value in C; this is a hermitian metric. Here, if N is a self adjoint integrable endomorphism on (M, g),

• ne may introduce:

h := νn−1g( · , · )+νn−2g( · , N · )+. . .+νg( · , Nn−2 · )+g( · , Nn−1 · ) with value in R[ν]; it is R[ν]-bilinear.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Another parallel with complex Calculus On a K¨ ahler manifold (M, J, g), one introduces: h := g( · , · ) + ig( · , J · ) with value in C; this is a hermitian metric. Here, if N is a self adjoint integrable endomorphism on (M, g),

• ne may introduce:

h := νn−1g( · , · )+νn−2g( · , N · )+. . .+νg( · , Nn−2 · )+g( · , Nn−1 · ) with value in R[ν]; it is R[ν]-bilinear.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Theorem B A metric g is in GN if and only if the associated R[ν]-valued, R[ν]-bilinear metric h is nilomorphic. Corollary of Theorems A and B Applying Theorem A gives the expression of such an h appearing in Theorem B (polynomial expansion . . . ). We deduce the expression

• f g in real terms (which is more complicated than that of h).

Remark The obtained holonomy group is, generically, the commutant of N in SO(g).

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Theorem B A metric g is in GN if and only if the associated R[ν]-valued, R[ν]-bilinear metric h is nilomorphic. Corollary of Theorems A and B Applying Theorem A gives the expression of such an h appearing in Theorem B (polynomial expansion . . . ). We deduce the expression

• f g in real terms (which is more complicated than that of h).

Remark The obtained holonomy group is, generically, the commutant of N in SO(g).

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Theorem B A metric g is in GN if and only if the associated R[ν]-valued, R[ν]-bilinear metric h is nilomorphic. Corollary of Theorems A and B Applying Theorem A gives the expression of such an h appearing in Theorem B (polynomial expansion . . . ). We deduce the expression

• f g in real terms (which is more complicated than that of h).

Remark The obtained holonomy group is, generically, the commutant of N in SO(g).

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Theorem C You may reproduce theorem B for each possible form of As, to

• btain metrics such that:

A = As ⋉ N, with N in the commutant of As. It suffices to replace verywhere real functions by R[ν]-valued, nilomorphic ones, and then to consider the ”real part” of the

• btained metrics.

Let us understand this on an example.

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Theorem C, an example. Recall that g is a K¨ ahler metric if and

• nly if it is the real part of a sesquilinear metric that reads:

g ∂ ∂zj , ∂ ∂zk

• =

∂2u ∂zj∂zk , with u a real function. Now g is a K¨ ahler metric making some C-linear nilpotent endomorphism parallel if and only if it is the ”real part” of a C[ν]-valued, R[ν]-bilinear, C-sesquilinear metric h that reads: h ∂ ∂zj , ∂ ∂zk

• =

∂2u ∂zj∂zk , with u an R[ν]-valued function. Remark The obtained holonomy group is, generically, the commutant of N in U(J, g).

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Theorem C, an example. Recall that g is a K¨ ahler metric if and

• nly if it is the real part of a sesquilinear metric that reads:

g ∂ ∂zj , ∂ ∂zk

• =

∂2u ∂zj∂zk , with u a real function. Now g is a K¨ ahler metric making some C-linear nilpotent endomorphism parallel if and only if it is the ”real part” of a C[ν]-valued, R[ν]-bilinear, C-sesquilinear metric h that reads: h ∂ ∂zj , ∂ ∂zk

• =

∂2u ∂zj∂zk , with u an R[ν]-valued function. Remark The obtained holonomy group is, generically, the commutant of N in U(J, g).

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo

Motivation: the title term by term Results

Theorem C, an example. Recall that g is a K¨ ahler metric if and

• nly if it is the real part of a sesquilinear metric that reads:

g ∂ ∂zj , ∂ ∂zk

• =

∂2u ∂zj∂zk , with u a real function. Now g is a K¨ ahler metric making some C-linear nilpotent endomorphism parallel if and only if it is the ”real part” of a C[ν]-valued, R[ν]-bilinear, C-sesquilinear metric h that reads: h ∂ ∂zj , ∂ ∂zk

• =

∂2u ∂zj∂zk , with u an R[ν]-valued function. Remark The obtained holonomy group is, generically, the commutant of N in U(J, g).

Charles Boubel Universit´ e de Strasbourg The algebra of the parallel endomorphisms of a germ of pseudo