Invariants of Diffeomorphism Groups Actions Valentin Lychagin and - PowerPoint PPT Presentation

D 0 = e ; D 1 = GL ( TM ) . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

D 0 = e ; D 1 = GL ( TM ) . For k ≥ 2 0 → S k T ∗ ⊗ T → D k → D k − 1 → e Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

D 0 = e ; D 1 = GL ( TM ) . For k ≥ 2 0 → S k T ∗ ⊗ T → D k → D k − 1 → e In words: D k is an extension of D k − 1 by Abelian group S k T ∗ ⊗ T . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Quadratic differential forms Let q : S 2 T ∗ M → M be the symmetric power of the cotangent bundle. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Quadratic differential forms Let q : S 2 T ∗ M → M be the symmetric power of the cotangent bundle. Smooth sections C ∞ ( q ) are quadratic differential forms on M . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Quadratic differential forms Let q : S 2 T ∗ M → M be the symmetric power of the cotangent bundle. Smooth sections C ∞ ( q ) are quadratic differential forms on M . Let J k ( q ) be the manifold of k -jets of quadratic differential forms, and q k : J k ( q ) → M the bundles of jets. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Quadratic differential forms Let q : S 2 T ∗ M → M be the symmetric power of the cotangent bundle. Smooth sections C ∞ ( q ) are quadratic differential forms on M . Let J k ( q ) be the manifold of k -jets of quadratic differential forms, and q k : J k ( q ) → M the bundles of jets. Exact sequences of vector bundles 0 → S k T ∗ ⊗ S 2 T ∗ → J k ( q ) → J k − 1 ( q ) → 0 . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Actions φ : M → M - local diffeomorphism, g - is a quadratic differential form: φ : g �→ φ ( g ) = φ ∗− 1 ( g ) the natural action. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Actions φ : M → M - local diffeomorphism, g - is a quadratic differential form: φ : g �→ φ ( g ) = φ ∗− 1 ( g ) the natural action. Jet -level. Let φ ( a ) = a , [ φ ] k + 1 ∈ D k + 1 ( k + 1 ) -jet of φ at the a point a , and [ g ] k a ∈ J k a ( q ) k -jet of g at the point a . Then [ φ ] k + 1 : [ g ] k a �→ [ φ ( g )] k a a defines D k + 1 -action on J k a ( q ) . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Important algebraic structure For any smooth bundle π : E ( π ) → M fibres of projections π k , 0 : J k ( π ) → J 0 ( π ) = E ( π ) are algebraic manifolds wrt canonical jet-coordinates. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Important algebraic structure For any smooth bundle π : E ( π ) → M fibres of projections π k , 0 : J k ( π ) → J 0 ( π ) = E ( π ) are algebraic manifolds wrt canonical jet-coordinates. The structure is natural in the sence that smooth bundle automorphisms induce birational isomorphisms of the algebraic manifolds. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Important algebraic structure For any smooth bundle π : E ( π ) → M fibres of projections π k , 0 : J k ( π ) → J 0 ( π ) = E ( π ) are algebraic manifolds wrt canonical jet-coordinates. The structure is natural in the sence that smooth bundle automorphisms induce birational isomorphisms of the algebraic manifolds. Important: Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Important algebraic structure For any smooth bundle π : E ( π ) → M fibres of projections π k , 0 : J k ( π ) → J 0 ( π ) = E ( π ) are algebraic manifolds wrt canonical jet-coordinates. The structure is natural in the sence that smooth bundle automorphisms induce birational isomorphisms of the algebraic manifolds. Important: J k a ( q ) - algebraic manifolds, Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Important algebraic structure For any smooth bundle π : E ( π ) → M fibres of projections π k , 0 : J k ( π ) → J 0 ( π ) = E ( π ) are algebraic manifolds wrt canonical jet-coordinates. The structure is natural in the sence that smooth bundle automorphisms induce birational isomorphisms of the algebraic manifolds. Important: J k a ( q ) - algebraic manifolds, D k + 1 - algebraic groups, Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Important algebraic structure For any smooth bundle π : E ( π ) → M fibres of projections π k , 0 : J k ( π ) → J 0 ( π ) = E ( π ) are algebraic manifolds wrt canonical jet-coordinates. The structure is natural in the sence that smooth bundle automorphisms induce birational isomorphisms of the algebraic manifolds. Important: J k a ( q ) - algebraic manifolds, D k + 1 - algebraic groups, D k + 1 × J k a ( q ) → J k a ( q ) -algebraic actions. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Rational differential invariants Rosenlicht theorem = ⇒ rational invariants of the D k + 1 -action on J k a ( q ) separate regular orbits. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Rational differential invariants Rosenlicht theorem = ⇒ rational invariants of the D k + 1 -action on J k a ( q ) separate regular orbits. We call the rational D k + 1 − invariants metric invariants . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Rational differential invariants Rosenlicht theorem = ⇒ rational invariants of the D k + 1 -action on J k a ( q ) separate regular orbits. We call the rational D k + 1 − invariants metric invariants . Let F k the field of metric invariants, then trdeg ( F k ) = codim ( regular D k + 1 − orbit ) . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Rational differential invariants Rosenlicht theorem = ⇒ rational invariants of the D k + 1 -action on J k a ( q ) separate regular orbits. We call the rational D k + 1 − invariants metric invariants . Let F k the field of metric invariants, then trdeg ( F k ) = codim ( regular D k + 1 − orbit ) . Metric Hilbert function : k �→ H ( k ) = trdeg ( F k ) − trdeg ( F k − 1 ) the "number of metric invariants of pure order k " . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Rational differential invariants Rosenlicht theorem = ⇒ rational invariants of the D k + 1 -action on J k a ( q ) separate regular orbits. We call the rational D k + 1 − invariants metric invariants . Let F k the field of metric invariants, then trdeg ( F k ) = codim ( regular D k + 1 − orbit ) . Metric Hilbert function : k �→ H ( k ) = trdeg ( F k ) − trdeg ( F k − 1 ) the "number of metric invariants of pure order k " . Metric Poincaré function : Π ( t ) = ∑ H ( k ) t k . k ≥ 0 Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Orbits Denote by ζ k the following operator sym ζ k : S k + 1 T ∗ ⊗ T → S k T ∗ ⊗ T ∗ ⊗ T → S k T ∗ ⊗ S 2 T ∗ , δ where δ is the Spencer δ -operator. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Orbits Denote by ζ k the following operator sym ζ k : S k + 1 T ∗ ⊗ T → S k T ∗ ⊗ T ∗ ⊗ T → S k T ∗ ⊗ S 2 T ∗ , δ where δ is the Spencer δ -operator. The tangent space T θ ( ∆ k + 1 θ ) to the orbit ∆ k + 1 θ , θ ∈ J k a ( q ) coinside with the image of ζ k : T θ ( ∆ k + 1 θ ) = Im ζ k . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Orbits Denote by ζ k the following operator sym ζ k : S k + 1 T ∗ ⊗ T → S k T ∗ ⊗ T ∗ ⊗ T → S k T ∗ ⊗ S 2 T ∗ , δ where δ is the Spencer δ -operator. The tangent space T θ ( ∆ k + 1 θ ) to the orbit ∆ k + 1 θ , θ ∈ J k a ( q ) coinside with the image of ζ k : T θ ( ∆ k + 1 θ ) = Im ζ k . The normal space to the orbit ∆ k + 1 θ is Coker ζ k . We call Coker ζ k , k ≥ 2 , space of curvature tensors of order k . Coker ζ 2 is the space of so-called algebraic curvature tensors. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Orbits Denote by ζ k the following operator sym ζ k : S k + 1 T ∗ ⊗ T → S k T ∗ ⊗ T ∗ ⊗ T → S k T ∗ ⊗ S 2 T ∗ , δ where δ is the Spencer δ -operator. The tangent space T θ ( ∆ k + 1 θ ) to the orbit ∆ k + 1 θ , θ ∈ J k a ( q ) coinside with the image of ζ k : T θ ( ∆ k + 1 θ ) = Im ζ k . The normal space to the orbit ∆ k + 1 θ is Coker ζ k . We call Coker ζ k , k ≥ 2 , space of curvature tensors of order k . Coker ζ 2 is the space of so-called algebraic curvature tensors. Ker ζ k = 0 , when k ≥ 1 and n = dim M ≥ 3 . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Invariants Computing dimensions of Coker ζ k leads us to the following formulae for metric Hilbert function. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Invariants Computing dimensions of Coker ζ k leads us to the following formulae for metric Hilbert function. The Hilbert function of metric invariants is: n ( n + 3 ) ( n − 1 ) ( n − 2 ) H n ( 2 ) = , 12 � n + k − 1 � n ( k − 1 ) = H n ( k ) , 2 k + 1 when n = dim M ≥ 3 and k ≥ 3 . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Invariants Computing dimensions of Coker ζ k leads us to the following formulae for metric Hilbert function. The Hilbert function of metric invariants is: n ( n + 3 ) ( n − 1 ) ( n − 2 ) H n ( 2 ) = , 12 � n + k − 1 � n ( k − 1 ) = H n ( k ) , 2 k + 1 when n = dim M ≥ 3 and k ≥ 3 . In the case dim M = 2 , H 2 ( 2 ) = H 2 ( 3 ) = 1 , H 2 ( k ) = k − 1 , when k ≥ 4 . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Invariants Computing dimensions of Coker ζ k leads us to the following formulae for metric Hilbert function. The Hilbert function of metric invariants is: n ( n + 3 ) ( n − 1 ) ( n − 2 ) H n ( 2 ) = , 12 � n + k − 1 � n ( k − 1 ) = H n ( k ) , 2 k + 1 when n = dim M ≥ 3 and k ≥ 3 . In the case dim M = 2 , H 2 ( 2 ) = H 2 ( 3 ) = 1 , H 2 ( k ) = k − 1 , when k ≥ 4 . The above formulae have been obtained by Zorawski, K. (1892), for the case n = 2 , and by Haskins, C.N. (1902), for the case n ≥ 3. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

For small dimensions ( n ≤ 4) and low orders ( k ≤ 9 ) , we have the following table of values of the Hilbert function: n \ k 2 3 4 5 6 7 8 9 2 1 1 3 4 5 6 7 8 3 3 15 27 42 60 71 105 132 4 14 60 126 224 360 540 770 1056 Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Tresse derivations Total differential . For f ∈ C ∞ � � we denote by � J k π df the total differential of f : n df � ∑ df = dx i , dx i i = 1 d where dx i total derivatives. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Tresse derivations Total differential . For f ∈ C ∞ � � we denote by � J k π df the total differential of f : n df � ∑ df = dx i , dx i i = 1 d where dx i total derivatives. We say that functions f 1 , ..., f n ∈ C ∞ � � J k π , where n = dim M , are in general position if � df 1 ∧ · · · ∧ � df n � = 0 in an open domain. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Tresse derivations For functions f 1 , ..., f n in general position and arbitrary f ∈ C ∞ � � J k π we have � df = λ 1 � df 1 + · · · + λ n � df n in the domain of definition. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Tresse derivations For functions f 1 , ..., f n in general position and arbitrary f ∈ C ∞ � � J k π we have � df = λ 1 � df 1 + · · · + λ n � df n in the domain of definition. Functions λ 1 , ..., λ n ∈ C ∞ � � J k + 1 π are said to be Tresse derivatives of f and will be denoted Df , ...., Df . Df 1 Df n Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Tresse derivations For functions f 1 , ..., f n in general position and arbitrary f ∈ C ∞ � � J k π we have � df = λ 1 � df 1 + · · · + λ n � df n in the domain of definition. Functions λ 1 , ..., λ n ∈ C ∞ � � J k + 1 π are said to be Tresse derivatives of f and will be denoted Df , ...., Df . Df 1 Df n Important observation : If f 1 , ..., f n are metric invariants of order k then Df Df i are metric invariants too for any metric invariant f . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Tresse derivations For functions f 1 , ..., f n in general position and arbitrary f ∈ C ∞ � � J k π we have � df = λ 1 � df 1 + · · · + λ n � df n in the domain of definition. Functions λ 1 , ..., λ n ∈ C ∞ � � J k + 1 π are said to be Tresse derivatives of f and will be denoted Df , ...., Df . Df 1 Df n Important observation : If f 1 , ..., f n are metric invariants of order k then Df Df i are metric invariants too for any metric invariant f . Remark that order of Df Df i , as a rule, higher then order of f . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Data Formally integrable algebraic PDEs system E k ⊂ J k π . That is, fibres of projection π k , 0 : E k → J 0 π are irreducible algebraic manifolds. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Data Formally integrable algebraic PDEs system E k ⊂ J k π . That is, fibres of projection π k , 0 : E k → J 0 π are irreducible algebraic manifolds. Algebraic Lie pseudo group G of symmetries of E , acting in a transitive way on J 0 π . That is, a Lie pseudogroup given by algebraic differential equations. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Data Formally integrable algebraic PDEs system E k ⊂ J k π . That is, fibres of projection π k , 0 : E k → J 0 π are irreducible algebraic manifolds. Algebraic Lie pseudo group G of symmetries of E , acting in a transitive way on J 0 π . That is, a Lie pseudogroup given by algebraic differential equations. By rational differerential G -invariants of order ≤ l we mean fibrewise rational functions on prolongation E l = E ( l − k ) ⊂ J l π which are k G -invariant. Denote by F l the field of invariants of order ≤ l . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Statement There are basic invariants : Q 1 , ..., Q n , P 1 , ..., P m , such that any rational differential G -invariant is a rational function of Q , P and the Tresse derivatives D | σ | P DQ σ . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Statement There are basic invariants : Q 1 , ..., Q n , P 1 , ..., P m , such that any rational differential G -invariant is a rational function of Q , P and the Tresse derivatives D | σ | P DQ σ . Rational differential G -invariants separate regular orbits. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Statement There are basic invariants : Q 1 , ..., Q n , P 1 , ..., P m , such that any rational differential G -invariant is a rational function of Q , P and the Tresse derivatives D | σ | P DQ σ . Rational differential G -invariants separate regular orbits. trdeg ( F l ) = codim ( regular G - orbit in E l ) . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Application: metric invariants n = 2 . Basic invariants: Q 1 = k g , Q 2 = g ( dk g , dk g ) , P 1 , P 2 , P 3 , where g = P 1 � dQ 2 1 + 2 P 2 � dQ 1 · � dQ 2 + P 3 � dQ 2 2 and k g is the curvature of g . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Application: metric invariants n = 2 . Basic invariants: Q 1 = k g , Q 2 = g ( dk g , dk g ) , P 1 , P 2 , P 3 , where g = P 1 � dQ 2 1 + 2 P 2 � dQ 1 · � dQ 2 + P 3 � dQ 2 2 and k g is the curvature of g . Remark that Q 1 is an invariant of order 2 , Q 2 -order 3 , and P 1 , P 2 , P 3 have order 4 . All of them rational. Therefore, due to Rosenlicht theorem and Zorawski result (see above) they do generate the field of metric invariants of order 4 . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Application: metric invariants n = 2 . Basic invariants: Q 1 = k g , Q 2 = g ( dk g , dk g ) , P 1 , P 2 , P 3 , where g = P 1 � dQ 2 1 + 2 P 2 � dQ 1 · � dQ 2 + P 3 � dQ 2 2 and k g is the curvature of g . Remark that Q 1 is an invariant of order 2 , Q 2 -order 3 , and P 1 , P 2 , P 3 have order 4 . All of them rational. Therefore, due to Rosenlicht theorem and Zorawski result (see above) they do generate the field of metric invariants of order 4 . Lie-Tresse theorem states that all metric invariants are rational functions of Q , P and Tresse derivatives D | σ | P DQ σ . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Lie -Tresse Theorem Application: metric invariants n = 2 . Basic invariants: Q 1 = k g , Q 2 = g ( dk g , dk g ) , P 1 , P 2 , P 3 , where g = P 1 � dQ 2 1 + 2 P 2 � dQ 1 · � dQ 2 + P 3 � dQ 2 2 and k g is the curvature of g . Remark that Q 1 is an invariant of order 2 , Q 2 -order 3 , and P 1 , P 2 , P 3 have order 4 . All of them rational. Therefore, due to Rosenlicht theorem and Zorawski result (see above) they do generate the field of metric invariants of order 4 . Lie-Tresse theorem states that all metric invariants are rational functions of Q , P and Tresse derivatives D | σ | P DQ σ . Singularity condition has order 4 : Σ 4 = { � dQ 1 ∧ � dQ 2 = 0 } ⊂ J 4 q . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Invariants n ≥ 3 . Basic invariants: Tr Ric g , Q 2 = Tr Ric 2 g , ..., Q n = Tr Ric n Q 1 = g , P ij , where g = ∑ P i , j � dQ i · � dQ j , i , j and Ric g : T → T the Ricci operator. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Invariants n ≥ 3 . Basic invariants: Tr Ric g , Q 2 = Tr Ric 2 g , ..., Q n = Tr Ric n Q 1 = g , P ij , where g = ∑ P i , j � dQ i · � dQ j , i , j and Ric g : T → T the Ricci operator. Remark that Q i are invariants of order 2 , and P ij have order 3 . Obviously, they together with the Tresse derivatives generate the field of metric invariants. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Invariants n ≥ 3 . Basic invariants: Tr Ric g , Q 2 = Tr Ric 2 g , ..., Q n = Tr Ric n Q 1 = g , P ij , where g = ∑ P i , j � dQ i · � dQ j , i , j and Ric g : T → T the Ricci operator. Remark that Q i are invariants of order 2 , and P ij have order 3 . Obviously, they together with the Tresse derivatives generate the field of metric invariants. Singularity condition has order 3 : Σ 4 = { � dQ 1 ∧ ... ∧ � dQ n = 0 } ⊂ J 3 q . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Factor Equation Let n = 2 , and let ( M , g ) be a fixed two-dimensional Riemannian manifold, and let Σ g ⊂ M the set of singular points of g , (i.e. j 4 ( g ) ∈ Σ 4 ). Assume that Σ g � = M . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Factor Equation Let n = 2 , and let ( M , g ) be a fixed two-dimensional Riemannian manifold, and let Σ g ⊂ M the set of singular points of g , (i.e. j 4 ( g ) ∈ Σ 4 ). Assume that Σ g � = M . Take the plane R 2 with fixed coordinates ( x 1 , x 2 ) and consider the following map ( invariantization ): I : M → R 2 , where x 1 = Q 1 ( g ) , x 2 = Q 2 ( g ) . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Factor Equation Let n = 2 , and let ( M , g ) be a fixed two-dimensional Riemannian manifold, and let Σ g ⊂ M the set of singular points of g , (i.e. j 4 ( g ) ∈ Σ 4 ). Assume that Σ g � = M . Take the plane R 2 with fixed coordinates ( x 1 , x 2 ) and consider the following map ( invariantization ): I : M → R 2 , where x 1 = Q 1 ( g ) , x 2 = Q 2 ( g ) . At regular points M � Σ g this mapping I is a covering over a domain D g ⊂ R 2 : I : M � Σ g → D g ⊂ R 2 . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Factor Equation Write down metric I − 1 ∗ ( g ) in invariant coordinates h = P 1 ( g ) dx 2 1 + 2 P 2 ( g ) dx 1 · dx 2 + P 3 ( g ) dx 2 2 . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Factor Equation Write down metric I − 1 ∗ ( g ) in invariant coordinates h = P 1 ( g ) dx 2 1 + 2 P 2 ( g ) dx 1 · dx 2 + P 3 ( g ) dx 2 2 . Then the metric h is invariantly related to g and satisfies the following factor equation E metric : Curv ( h ) = x 1 , h ( dx 1 , dx 1 ) = x 2 . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Factor Equation Write down metric I − 1 ∗ ( g ) in invariant coordinates h = P 1 ( g ) dx 2 1 + 2 P 2 ( g ) dx 1 · dx 2 + P 3 ( g ) dx 2 2 . Then the metric h is invariantly related to g and satisfies the following factor equation E metric : Curv ( h ) = x 1 , h ( dx 1 , dx 1 ) = x 2 . Equivalence classes of metrics ⇐ ⇒ Solutions of PDEs system E metric . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Example We say that a Riemann surface ( M , g ) is orthogonal if the net Q 1 ( g ) = constant , Q 2 ( g ) = constant , is orthogonal, i.e. � � �� g ( � dk , � dk , � � d g dk = 0 . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Example We say that a Riemann surface ( M , g ) is orthogonal if the net Q 1 ( g ) = constant , Q 2 ( g ) = constant , is orthogonal, i.e. � � �� g ( � dk , � � dk , � d g dk = 0 . This equation is natural, and the factor equation for orthogonal surfaces has the form y 4 c 2 c xx + yc y + 2 c + xy 2 c 3 = 0 2 � x , y 2 � in coordinates ( Q 1 , Q 2 ) ⇔ . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Example We say that a Riemann surface ( M , g ) is orthogonal if the net Q 1 ( g ) = constant , Q 2 ( g ) = constant , is orthogonal, i.e. � � �� g ( � dk , � � dk , � d g dk = 0 . This equation is natural, and the factor equation for orthogonal surfaces has the form y 4 c 2 c xx + yc y + 2 c + xy 2 c 3 = 0 2 � x , y 2 � in coordinates ( Q 1 , Q 2 ) ⇔ . Orthogonal surfaces ⇐ ⇒ solutions of the above equation Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Example We say that a Riemann surface ( M , g ) is orthogonal if the net Q 1 ( g ) = constant , Q 2 ( g ) = constant , is orthogonal, i.e. � � �� g ( � dk , � � dk , � d g dk = 0 . This equation is natural, and the factor equation for orthogonal surfaces has the form y 4 c 2 c xx + yc y + 2 c + xy 2 c 3 = 0 2 � x , y 2 � in coordinates ( Q 1 , Q 2 ) ⇔ . Orthogonal surfaces ⇐ ⇒ solutions of the above equation The factor equation has the scale symmetries � � e 2 t x , e 3 t y , e − 4 t c ( x , y , c ) �→ , which generate the Bäcklund type transformations on the class of orthogonal surfaces. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Factor Equation Let n ≥ 3 , and let ( M , g ) be a fixed n -dimensional Riemannian manifold, and let Σ g ⊂ M the set of Ricci singular points of g , (i.e. j 3 ( g ) ∈ Σ 3 ). Assume that Σ g � = M . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Factor Equation Let n ≥ 3 , and let ( M , g ) be a fixed n -dimensional Riemannian manifold, and let Σ g ⊂ M the set of Ricci singular points of g , (i.e. j 3 ( g ) ∈ Σ 3 ). Assume that Σ g � = M . Take space R n with fixed coordinates ( x 1 , ..., x n ) and consider the invariantization mapping : I : M → R n , where x 1 = Q 1 ( g ) , ..., x n = Q n ( g ) . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Factor Equation Let n ≥ 3 , and let ( M , g ) be a fixed n -dimensional Riemannian manifold, and let Σ g ⊂ M the set of Ricci singular points of g , (i.e. j 3 ( g ) ∈ Σ 3 ). Assume that Σ g � = M . Take space R n with fixed coordinates ( x 1 , ..., x n ) and consider the invariantization mapping : I : M → R n , where x 1 = Q 1 ( g ) , ..., x n = Q n ( g ) . At regular points M � Σ g this mapping I is a covering over a domain D g ⊂ R n : I : M � Σ g → D g ⊂ R n . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Factor Equation The factor equation E metric is the following equation on a metric h on R n : Tr Ric h = x 1 , Tr Ric 2 h = x 2 , ..., Tr Ric n h = x n . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Metric Factor Equation The factor equation E metric is the following equation on a metric h on R n : Tr Ric h = x 1 , Tr Ric 2 h = x 2 , ..., Tr Ric n h = x n . Equivalence classes of metrics ⇐ ⇒ Solutions of PDEs system E metric . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Einstein equation Let M be an oriented 4-dimensional manifold, g be a Lorentzian metric on M . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Einstein equation Let M be an oriented 4-dimensional manifold, g be a Lorentzian metric on M . The vacuum Einstein equation G µν = 0 where G µν = R µν − R 2 g µν is the Einstein tensor, with R µν -Ricci curvature tensor and R the scalar curvature tensor. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Einstein equation Let M be an oriented 4-dimensional manifold, g be a Lorentzian metric on M . The vacuum Einstein equation G µν = 0 where G µν = R µν − R 2 g µν is the Einstein tensor, with R µν -Ricci curvature tensor and R the scalar curvature tensor. Let E 2 ⊂ J 2 q be a submanifold corresponding to the equation and let E k = E ( k − 2 ) ⊂ J k q 2 be prolongations of the Einstein equation. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Relativistic Invariants Important remarks: Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Relativistic Invariants Important remarks: The Einstein equation is natural, i.e. invariant of the prolonged action 1 of the diffeomorphism group, and Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Relativistic Invariants Important remarks: The Einstein equation is natural, i.e. invariant of the prolonged action 1 of the diffeomorphism group, and Formally integrable. 2 Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Relativistic Invariants Important remarks: The Einstein equation is natural, i.e. invariant of the prolonged action 1 of the diffeomorphism group, and Formally integrable. 2 Fibres of the projections 3 π k : E k → M are irreducible algebraic manifolds and actions of the differential groups on these fibres are algebraic. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Relativistic Invariants Important remarks: The Einstein equation is natural, i.e. invariant of the prolonged action 1 of the diffeomorphism group, and Formally integrable. 2 Fibres of the projections 3 π k : E k → M are irreducible algebraic manifolds and actions of the differential groups on these fibres are algebraic. A fibrewise rational function on the bundle π k : E k → M we call relativistic invariant of order ≤ k if the function is invariant with respect to the prolonged action of the diffeomorphism group. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Weyl Operator The Hodge operator ∗ : Λ 2 T ∗ → Λ 2 T ∗ defines the comlex structure in the bundle of differential 2-forms. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Weyl Operator The Hodge operator ∗ : Λ 2 T ∗ → Λ 2 T ∗ defines the comlex structure in the bundle of differential 2-forms. The Weyl tensor, considered as operator, defines Weyl operator W g : Λ 2 T ∗ → Λ 2 T ∗ , which Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Weyl Operator The Hodge operator ∗ : Λ 2 T ∗ → Λ 2 T ∗ defines the comlex structure in the bundle of differential 2-forms. The Weyl tensor, considered as operator, defines Weyl operator W g : Λ 2 T ∗ → Λ 2 T ∗ , which commutes with ∗ , i.e. W g is a C -linear operator. Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Weyl Operator The Hodge operator ∗ : Λ 2 T ∗ → Λ 2 T ∗ defines the comlex structure in the bundle of differential 2-forms. The Weyl tensor, considered as operator, defines Weyl operator W g : Λ 2 T ∗ → Λ 2 T ∗ , which commutes with ∗ , i.e. W g is a C -linear operator. Tr C W g = 0 . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Weyl Operator The Hodge operator ∗ : Λ 2 T ∗ → Λ 2 T ∗ defines the comlex structure in the bundle of differential 2-forms. The Weyl tensor, considered as operator, defines Weyl operator W g : Λ 2 T ∗ → Λ 2 T ∗ , which commutes with ∗ , i.e. W g is a C -linear operator. Tr C W g = 0 . Let √ Tr C W 2 = I 1 , g + − 1 I 2 , g , g √ Tr C W 3 = I 3 , g + − 1 I 4 , g . g Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Basic Relativistics invariants Denote by I 1 , I 2 , I 3 , I 4 functions on E 2 corresponding to I i , g . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Basic Relativistics invariants Denote by I 1 , I 2 , I 3 , I 4 functions on E 2 corresponding to I i , g . They are rational and invariant under diffeomorphisms Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39

Basic Relativistics invariants Denote by I 1 , I 2 , I 3 , I 4 functions on E 2 corresponding to I i , g . They are rational and invariant under diffeomorphisms dim(Weyl tensors)-dim ( so ( 1 , 3 )) = 10 − 6 = 4 . Workshop on “Infinite-dimensional Riemannia Lychagin & Yumaguzhin (University of Tromso) Differential Invariants / 39