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( T M ). So at each point p ∈ M , U | p ∈ End( T p M ). 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( T M ). So at each point p ∈ M , U | p ∈ End( T p M ). 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( T M ). So at each point p ∈ M , U | p ∈ End( T p M ). 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( T M )); DU = 0 } , equivalently A := { U ∈ End( T p M ); ∀ h ∈ H p , 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( T M )); DU = 0 } , equivalently A := { U ∈ End( T p M ); ∀ h ∈ H p , 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 = � J 1 , J 2 , J 3 � ≃ H (hyperk¨ ahler metrics). – because of a theorem of de Rham: If a subspace E p of T p M is H p -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 ) ≃ ( E p , g 1 ) × ( E ⊥ p , g 2 ) corresponding to a decomposition H = H 1 × H 2 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 = � J 1 , J 2 , J 3 � ≃ H (hyperk¨ ahler metrics). – because of a theorem of de Rham: If a subspace E p of T p M is H p -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 ) ≃ ( E p , g 1 ) × ( E ⊥ p , g 2 ) corresponding to a decomposition H = H 1 × H 2 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 = � J 1 , J 2 , J 3 � ≃ H (hyperk¨ ahler metrics). – because of a theorem of de Rham: If a subspace E p of T p M is H p -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 ) ≃ ( E p , g 1 ) × ( E ⊥ p , g 2 ) corresponding to a decomposition H = H 1 × H 2 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 = � J 1 , J 2 , J 3 � ≃ H (hyperk¨ ahler metrics). – because of a theorem of de Rham: If a subspace E p of T p M is H p -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 ) ≃ ( E p , g 1 ) × ( E ⊥ p , g 2 ) corresponding to a decomposition H = H 1 × H 2 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 = � J 1 , J 2 , J 3 � ≃ H (hyperk¨ ahler metrics). – because of a theorem of de Rham: If a subspace E p of T p M is H p -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 ) ≃ ( E p , g 1 ) × ( E ⊥ p , g 2 ) corresponding to a decomposition H = H 1 × H 2 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 H p acts irreducibly on T p M : 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 H p acts irreducibly on T p M : 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 H p acts irreducibly on T p M : 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 E p of T p M is H p -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 ) ≃ ( E p , g 1 ) × ( E ⊥ p , g 2 ) corresponding to a decomposition H = H 1 × H 2 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
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