Invariants of Diffeomorphism Groups Actions Valentin Lychagin and - - PowerPoint PPT Presentation
Invariants of Diffeomorphism Groups Actions Valentin Lychagin and Valery Yumaguzhin Russian Academy of Sciences, Russia & University of Troms, Norway Workshop on Infinite-dimensional Riemannian geometry January 12 16, 2015,
Valentin Lychagin and Valery Yumaguzhin
Russian Academy of Sciences, Russia & University of Tromsø, Norway
Workshop on “Infinite-dimensional Riemannian geometry ” January 12 — 16, 2015, Vienna, Austria
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
The field of rational differential invariants.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
The field of rational differential invariants. Hilbert and Poincare functions for metrics.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
The field of rational differential invariants. Hilbert and Poincare functions for metrics. Factor Equations and the equivalence problem for metrics.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
The field of rational differential invariants. Hilbert and Poincare functions for metrics. Factor Equations and the equivalence problem for metrics.
Vacuum Einstein equations.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
The field of rational differential invariants. Hilbert and Poincare functions for metrics. Factor Equations and the equivalence problem for metrics.
Vacuum Einstein equations.
The field of relativistic invariants.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
The field of rational differential invariants. Hilbert and Poincare functions for metrics. Factor Equations and the equivalence problem for metrics.
Vacuum Einstein equations.
The field of relativistic invariants. Hilbert and Poincare functions for relativistic invariants.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
The field of rational differential invariants. Hilbert and Poincare functions for metrics. Factor Equations and the equivalence problem for metrics.
Vacuum Einstein equations.
The field of relativistic invariants. Hilbert and Poincare functions for relativistic invariants. Factor Einstein Equations and the equivalence problem for Einstein metrics.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
The field of rational differential invariants. Hilbert and Poincare functions for metrics. Factor Equations and the equivalence problem for metrics.
Vacuum Einstein equations.
The field of relativistic invariants. Hilbert and Poincare functions for relativistic invariants. Factor Einstein Equations and the equivalence problem for Einstein metrics.
Einstein - Maxwell equations
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
The field of rational differential invariants. Hilbert and Poincare functions for metrics. Factor Equations and the equivalence problem for metrics.
Vacuum Einstein equations.
The field of relativistic invariants. Hilbert and Poincare functions for relativistic invariants. Factor Einstein Equations and the equivalence problem for Einstein metrics.
Einstein - Maxwell equations
The field of electromagnetic invariants.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
The field of rational differential invariants. Hilbert and Poincare functions for metrics. Factor Equations and the equivalence problem for metrics.
Vacuum Einstein equations.
The field of relativistic invariants. Hilbert and Poincare functions for relativistic invariants. Factor Einstein Equations and the equivalence problem for Einstein metrics.
Einstein - Maxwell equations
The field of electromagnetic invariants. Hilbert and Poincare functions for electromagnetic invariants.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Diffeomorphisms and Naturality Principle in Physics Metrics (and Gravity)
The field of rational differential invariants. Hilbert and Poincare functions for metrics. Factor Equations and the equivalence problem for metrics.
Vacuum Einstein equations.
The field of relativistic invariants. Hilbert and Poincare functions for relativistic invariants. Factor Einstein Equations and the equivalence problem for Einstein metrics.
Einstein - Maxwell equations
The field of electromagnetic invariants. Hilbert and Poincare functions for electromagnetic invariants. Factor Einstein-Maxwell Equations and the equivalence problem for solutions of Einstein - Maxwell equations.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
M is a smooth manifold.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
M is a smooth manifold. πk : Jk
0 (M, M) → M × M - the bundle of k-jets of local
diffeomorphisms (of M ).
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
M is a smooth manifold. πk : Jk
0 (M, M) → M × M - the bundle of k-jets of local
diffeomorphisms (of M ). The bundles πk,k−1 : Jk
0 (M, M) → Jk−1
(M, M) are affine, when k ≥ 2, and the associated vector bundles are SkT∗ ⊗ T.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
M is a smooth manifold. πk : Jk
0 (M, M) → M × M - the bundle of k-jets of local
diffeomorphisms (of M ). The bundles πk,k−1 : Jk
0 (M, M) → Jk−1
(M, M) are affine, when k ≥ 2, and the associated vector bundles are SkT∗ ⊗ T. Fix a ∈ M, then
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
M is a smooth manifold. πk : Jk
0 (M, M) → M × M - the bundle of k-jets of local
diffeomorphisms (of M ). The bundles πk,k−1 : Jk
0 (M, M) → Jk−1
(M, M) are affine, when k ≥ 2, and the associated vector bundles are SkT∗ ⊗ T. Fix a ∈ M, then
fibres Dk = π−1
k,0 (a, a)
are algebraic Lie groups, (differential group of order k )
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
M is a smooth manifold. πk : Jk
0 (M, M) → M × M - the bundle of k-jets of local
diffeomorphisms (of M ). The bundles πk,k−1 : Jk
0 (M, M) → Jk−1
(M, M) are affine, when k ≥ 2, and the associated vector bundles are SkT∗ ⊗ T. Fix a ∈ M, then
fibres Dk = π−1
k,0 (a, a)
are algebraic Lie groups, (differential group of order k ) projections πk,k−1 : Dk → Dk−1 are Lie group epimorphisms, and
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
M is a smooth manifold. πk : Jk
0 (M, M) → M × M - the bundle of k-jets of local
diffeomorphisms (of M ). The bundles πk,k−1 : Jk
0 (M, M) → Jk−1
(M, M) are affine, when k ≥ 2, and the associated vector bundles are SkT∗ ⊗ T. Fix a ∈ M, then
fibres Dk = π−1
k,0 (a, a)
are algebraic Lie groups, (differential group of order k ) projections πk,k−1 : Dk → Dk−1 are Lie group epimorphisms, and kernels ∆k = ker (πk,k−1 : Dk → Dk−1) are abelian Lie groups,and ∆k SkT∗ ⊗ T.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
D0 = e; D1 = GL ( TM) .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
D0 = e; D1 = GL ( TM) . For k ≥ 2 0 → SkT∗ ⊗ T →Dk → Dk−1 → e
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
D0 = e; D1 = GL ( TM) . For k ≥ 2 0 → SkT∗ ⊗ T →Dk → Dk−1 → e In words: Dk is an extension of Dk−1 by Abelian group SkT∗ ⊗ T.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let q : S2T∗M → M be the symmetric power of the cotangent bundle.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let q : S2T∗M → M be the symmetric power of the cotangent bundle. Smooth sections C ∞ (q) are quadratic differential forms on M.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let q : S2T∗M → M be the symmetric power of the cotangent bundle. Smooth sections C ∞ (q) are quadratic differential forms on M. Let Jk (q) be the manifold of k-jets of quadratic differential forms, and qk : Jk (q) → M the bundles of jets.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let q : S2T∗M → M be the symmetric power of the cotangent bundle. Smooth sections C ∞ (q) are quadratic differential forms on M. Let Jk (q) be the manifold of k-jets of quadratic differential forms, and qk : Jk (q) → M the bundles of jets. Exact sequences of vector bundles 0 → SkT∗ ⊗ S2T∗→ Jk (q) → Jk−1 (q) → 0.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
φ : M → M - local diffeomorphism, g - is a quadratic differential form: φ : g → φ (g) = φ∗−1 (g) the natural action.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
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
a
∈ Dk+1 (k + 1) -jet of φ at the point a, and [g]k
a ∈ Jk a (q) k -jet of g at the point a. Then
[φ]k+1
a
: [g]k
a → [φ(g)]k a
defines Dk+1 -action on Jk
a (q) .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
For any smooth bundle π : E (π) → M fibres of projections πk,0 : Jk (π) → J0 (π) = E (π) are algebraic manifolds wrt canonical jet-coordinates.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
For any smooth bundle π : E (π) → M fibres of projections πk,0 : Jk (π) → J0 (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.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
For any smooth bundle π : E (π) → M fibres of projections πk,0 : Jk (π) → J0 (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:
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
For any smooth bundle π : E (π) → M fibres of projections πk,0 : Jk (π) → J0 (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:
Jk
a (q) - algebraic manifolds,
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
For any smooth bundle π : E (π) → M fibres of projections πk,0 : Jk (π) → J0 (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:
Jk
a (q) - algebraic manifolds,
Dk+1 - algebraic groups,
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
For any smooth bundle π : E (π) → M fibres of projections πk,0 : Jk (π) → J0 (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:
Jk
a (q) - algebraic manifolds,
Dk+1 - algebraic groups, Dk+1 × Jk
a (q) → Jk a (q) -algebraic actions.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Rosenlicht theorem = ⇒ rational invariants of the Dk+1 -action on Jk
a (q) separate regular orbits.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Rosenlicht theorem = ⇒ rational invariants of the Dk+1 -action on Jk
a (q) separate regular orbits.
We call the rational Dk+1−invariants metric invariants.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Rosenlicht theorem = ⇒ rational invariants of the Dk+1 -action on Jk
a (q) separate regular orbits.
We call the rational Dk+1−invariants metric invariants. Let Fk the field of metric invariants, then trdeg (Fk) = codim (regular Dk+1 − orbit) .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Rosenlicht theorem = ⇒ rational invariants of the Dk+1 -action on Jk
a (q) separate regular orbits.
We call the rational Dk+1−invariants metric invariants. Let Fk the field of metric invariants, then trdeg (Fk) = codim (regular Dk+1 − orbit) . Metric Hilbert function: k → H(k) = trdeg (Fk) − trdeg (Fk−1) the "number of metric invariants of pure order k" .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Rosenlicht theorem = ⇒ rational invariants of the Dk+1 -action on Jk
a (q) separate regular orbits.
We call the rational Dk+1−invariants metric invariants. Let Fk the field of metric invariants, then trdeg (Fk) = codim (regular Dk+1 − orbit) . Metric Hilbert function: k → H(k) = trdeg (Fk) − trdeg (Fk−1) the "number of metric invariants of pure order k" . Metric Poincaré function: Π (t) = ∑
k≥0
H (k) tk.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by ζk the following operator ζk : Sk+1T∗ ⊗ T
δ
→ SkT∗ ⊗ T∗ ⊗ T
sym
→ SkT∗ ⊗ S2T∗, where δ is the Spencer δ-operator.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by ζk the following operator ζk : Sk+1T∗ ⊗ T
δ
→ SkT∗ ⊗ T∗ ⊗ T
sym
→ SkT∗ ⊗ S2T∗, where δ is the Spencer δ-operator. The tangent space Tθ (∆k+1θ) to the orbit ∆k+1θ, θ ∈ Jk
a (q)
coinside with the image of ζk : Tθ (∆k+1θ) = Im ζk.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by ζk the following operator ζk : Sk+1T∗ ⊗ T
δ
→ SkT∗ ⊗ T∗ ⊗ T
sym
→ SkT∗ ⊗ S2T∗, where δ is the Spencer δ-operator. The tangent space Tθ (∆k+1θ) to the orbit ∆k+1θ, θ ∈ Jk
a (q)
coinside with the image of ζk : Tθ (∆k+1θ) = Im ζk. The normal space to the orbit ∆k+1θ is Coker ζk. We call Coker ζk, k ≥ 2, space of curvature tensors of order k. Coker ζ2 is the space of so-called algebraic curvature tensors.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by ζk the following operator ζk : Sk+1T∗ ⊗ T
δ
→ SkT∗ ⊗ T∗ ⊗ T
sym
→ SkT∗ ⊗ S2T∗, where δ is the Spencer δ-operator. The tangent space Tθ (∆k+1θ) to the orbit ∆k+1θ, θ ∈ Jk
a (q)
coinside with the image of ζk : Tθ (∆k+1θ) = Im ζk. The normal space to the orbit ∆k+1θ is Coker ζk. We call Coker ζk, k ≥ 2, space of curvature tensors of order k. Coker ζ2 is the space of so-called algebraic curvature tensors. Ker ζk = 0, when k ≥ 1 and n = dim M ≥ 3.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Computing dimensions of Coker ζk leads us to the following formulae for metric Hilbert function.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Computing dimensions of Coker ζk leads us to the following formulae for metric Hilbert function. The Hilbert function of metric invariants is: Hn (2) = n (n + 3) (n − 1) (n − 2) 12 , Hn (k) = n (k − 1) 2 n + k − 1 k + 1
when n = dim M ≥ 3 and k ≥ 3.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Computing dimensions of Coker ζk leads us to the following formulae for metric Hilbert function. The Hilbert function of metric invariants is: Hn (2) = n (n + 3) (n − 1) (n − 2) 12 , Hn (k) = n (k − 1) 2 n + k − 1 k + 1
when n = dim M ≥ 3 and k ≥ 3. In the case dim M = 2, H2 (2) = H2 (3) = 1, H2 (k) = k − 1, when k ≥ 4.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Computing dimensions of Coker ζk leads us to the following formulae for metric Hilbert function. The Hilbert function of metric invariants is: Hn (2) = n (n + 3) (n − 1) (n − 2) 12 , Hn (k) = n (k − 1) 2 n + k − 1 k + 1
when n = dim M ≥ 3 and k ≥ 3. In the case dim M = 2, H2 (2) = H2 (3) = 1, H2 (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.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 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
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Tresse derivations
Total differential. For f ∈ C ∞ Jkπ
df the total differential of f :
n
i=1
df dxi dxi, where
d dxi total derivatives.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Tresse derivations
Total differential. For f ∈ C ∞ Jkπ
df the total differential of f :
n
i=1
df dxi dxi, where
d dxi total derivatives.
We say that functions f1, ..., fn ∈ C ∞ Jkπ
in general position if
dfn = 0 in an open domain.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Tresse derivations
For functions f1, ..., fn in general position and arbitrary f ∈ C ∞ Jkπ
df1 + · · · + λn dfn in the domain of definition.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Tresse derivations
For functions f1, ..., fn in general position and arbitrary f ∈ C ∞ Jkπ
df1 + · · · + λn dfn in the domain of definition. Functions λ1, ..., λn ∈ C ∞ Jk+1π
Df Df1 , ...., Df Dfn .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Tresse derivations
For functions f1, ..., fn in general position and arbitrary f ∈ C ∞ Jkπ
df1 + · · · + λn dfn in the domain of definition. Functions λ1, ..., λn ∈ C ∞ Jk+1π
Df Df1 , ...., Df Dfn . Important observation: If f1, ..., fn are metric invariants of order k then Df Dfi are metric invariants too for any metric invariant f .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Tresse derivations
For functions f1, ..., fn in general position and arbitrary f ∈ C ∞ Jkπ
df1 + · · · + λn dfn in the domain of definition. Functions λ1, ..., λn ∈ C ∞ Jk+1π
Df Df1 , ...., Df Dfn . Important observation: If f1, ..., fn are metric invariants of order k then Df Dfi are metric invariants too for any metric invariant f . Remark that order of Df
Dfi , as a rule, higher then order of f .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Data Formally integrable algebraic PDEs system Ek ⊂ Jkπ. That is, fibres of projection πk,0 : Ek → J0π are irreducible algebraic manifolds.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Data Formally integrable algebraic PDEs system Ek ⊂ Jkπ. That is, fibres of projection πk,0 : Ek → J0π are irreducible algebraic manifolds. Algebraic Lie pseudo group G of symmetries of E, acting in a transitive way on J0π. That is, a Lie pseudogroup given by algebraic differential equations.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Data Formally integrable algebraic PDEs system Ek ⊂ Jkπ. That is, fibres of projection πk,0 : Ek → J0π are irreducible algebraic manifolds. Algebraic Lie pseudo group G of symmetries of E, acting in a transitive way on J0π. 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 El = E(l−k)
k
⊂ Jlπ which are G-invariant. Denote by Fl the field of invariants of order ≤ l.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Statement
There are basic invariants: Q1, ..., Qn, P1, ..., Pm, such that any rational differential G-invariant is a rational function of Q, P and the Tresse derivatives D|σ|P DQσ .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Statement
There are basic invariants: Q1, ..., Qn, P1, ..., Pm, such that any rational differential G-invariant is a rational function of Q, P and the Tresse derivatives D|σ|P DQσ . Rational differential G-invariants separate regular orbits.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Statement
There are basic invariants: Q1, ..., Qn, P1, ..., Pm, such that any rational differential G-invariant is a rational function of Q, P and the Tresse derivatives D|σ|P DQσ . Rational differential G-invariants separate regular orbits. trdeg (Fl) = codim (regular G-orbit in El) .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Application: metric invariants
n = 2. Basic invariants: Q1 = kg, Q2 = g (dkg, dkg ) , P1, P2, P3, where g = P1 dQ2
1 + 2P2
dQ1 · dQ2 + P3 dQ2
2
and kg is the curvature of g.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Application: metric invariants
n = 2. Basic invariants: Q1 = kg, Q2 = g (dkg, dkg ) , P1, P2, P3, where g = P1 dQ2
1 + 2P2
dQ1 · dQ2 + P3 dQ2
2
and kg is the curvature of g. Remark that Q1 is an invariant of order 2, Q2 -order 3, and P1, P2, P3 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.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Application: metric invariants
n = 2. Basic invariants: Q1 = kg, Q2 = g (dkg, dkg ) , P1, P2, P3, where g = P1 dQ2
1 + 2P2
dQ1 · dQ2 + P3 dQ2
2
and kg is the curvature of g. Remark that Q1 is an invariant of order 2, Q2 -order 3, and P1, P2, P3 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 .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Application: metric invariants
n = 2. Basic invariants: Q1 = kg, Q2 = g (dkg, dkg ) , P1, P2, P3, where g = P1 dQ2
1 + 2P2
dQ1 · dQ2 + P3 dQ2
2
and kg is the curvature of g. Remark that Q1 is an invariant of order 2, Q2 -order 3, and P1, P2, P3 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 : Σ4 = { dQ1 ∧ dQ2 = 0} ⊂ J4q.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
n ≥ 3. Basic invariants: Q1 = Tr Ricg, Q2 = Tr Ric2
g, ..., Qn = Tr Ricn g,
Pij, where g = ∑
i,j
Pi,j dQi · dQj, and Ricg : T → T the Ricci operator.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
n ≥ 3. Basic invariants: Q1 = Tr Ricg, Q2 = Tr Ric2
g, ..., Qn = Tr Ricn g,
Pij, where g = ∑
i,j
Pi,j dQi · dQj, and Ricg : T → T the Ricci operator. Remark that Qi are invariants of order 2, and Pij have order 3. Obviously, they together with the Tresse derivatives generate the field
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
n ≥ 3. Basic invariants: Q1 = Tr Ricg, Q2 = Tr Ric2
g, ..., Qn = Tr Ricn g,
Pij, where g = ∑
i,j
Pi,j dQi · dQj, and Ricg : T → T the Ricci operator. Remark that Qi are invariants of order 2, and Pij have order 3. Obviously, they together with the Tresse derivatives generate the field
Singularity condition has order 3 : Σ4 = { dQ1 ∧ ... ∧ dQn = 0} ⊂ J3q.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let n = 2, and let (M, g) be a fixed two-dimensional Riemannian manifold, and let Σg ⊂ M the set of singular points of g, (i.e. j4 (g) ∈ Σ4 ). Assume that Σg = M.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let n = 2, and let (M, g) be a fixed two-dimensional Riemannian manifold, and let Σg ⊂ M the set of singular points of g, (i.e. j4 (g) ∈ Σ4 ). Assume that Σg = M. Take the plane R2 with fixed coordinates (x1, x2) and consider the following map (invariantization): I : M → R2, where x1 = Q1 (g) , x2 = Q2 (g) .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let n = 2, and let (M, g) be a fixed two-dimensional Riemannian manifold, and let Σg ⊂ M the set of singular points of g, (i.e. j4 (g) ∈ Σ4 ). Assume that Σg = M. Take the plane R2 with fixed coordinates (x1, x2) and consider the following map (invariantization): I : M → R2, where x1 = Q1 (g) , x2 = Q2 (g) . At regular points M Σg this mapping I is a covering over a domain Dg ⊂ R2 : I : M Σg → Dg ⊂ R2.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Write down metric I−1∗ (g) in invariant coordinates h = P1 (g) dx2
1 + 2P2 (g) dx1 · dx2 + P3 (g) dx2 2 .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Write down metric I−1∗ (g) in invariant coordinates h = P1 (g) dx2
1 + 2P2 (g) dx1 · dx2 + P3 (g) dx2 2 .
Then the metric h is invariantly related to g and satisfies the following factor equation Emetric: Curv (h) = x1, h (dx1, dx1) = x2.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Write down metric I−1∗ (g) in invariant coordinates h = P1 (g) dx2
1 + 2P2 (g) dx1 · dx2 + P3 (g) dx2 2 .
Then the metric h is invariantly related to g and satisfies the following factor equation Emetric: Curv (h) = x1, h (dx1, dx1) = x2. Equivalence classes of metrics ⇐ ⇒ Solutions of PDEs system Emetric.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
We say that a Riemann surface (M, g) is orthogonal if the net Q1 (g) = constant, Q2 (g) = constant, is orthogonal, i.e. g( dk, d
dk
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
We say that a Riemann surface (M, g) is orthogonal if the net Q1 (g) = constant, Q2 (g) = constant, is orthogonal, i.e. g( dk, d
dk
This equation is natural, and the factor equation for orthogonal surfaces has the form y4c2cxx + ycy + 2c + xy2c3 2 = 0 in coordinates (Q1, Q2) ⇔
.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
We say that a Riemann surface (M, g) is orthogonal if the net Q1 (g) = constant, Q2 (g) = constant, is orthogonal, i.e. g( dk, d
dk
This equation is natural, and the factor equation for orthogonal surfaces has the form y4c2cxx + ycy + 2c + xy2c3 2 = 0 in coordinates (Q1, Q2) ⇔
. Orthogonal surfaces ⇐ ⇒solutions of the above equation
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
We say that a Riemann surface (M, g) is orthogonal if the net Q1 (g) = constant, Q2 (g) = constant, is orthogonal, i.e. g( dk, d
dk
This equation is natural, and the factor equation for orthogonal surfaces has the form y4c2cxx + ycy + 2c + xy2c3 2 = 0 in coordinates (Q1, Q2) ⇔
. Orthogonal surfaces ⇐ ⇒solutions of the above equation The factor equation has the scale symmetries (x, y, c) →
which generate the Bäcklund type transformations on the class of
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let n ≥ 3, and let (M, g) be a fixed n-dimensional Riemannian manifold, and let Σg ⊂ M the set of Ricci singular points of g, (i.e. j3 (g) ∈ Σ3 ). Assume that Σg = M.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let n ≥ 3, and let (M, g) be a fixed n-dimensional Riemannian manifold, and let Σg ⊂ M the set of Ricci singular points of g, (i.e. j3 (g) ∈ Σ3 ). Assume that Σg = M. Take space Rn with fixed coordinates (x1, ..., xn) and consider the invariantization mapping: I : M → Rn, where x1 = Q1 (g) , ..., xn = Qn (g) .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let n ≥ 3, and let (M, g) be a fixed n-dimensional Riemannian manifold, and let Σg ⊂ M the set of Ricci singular points of g, (i.e. j3 (g) ∈ Σ3 ). Assume that Σg = M. Take space Rn with fixed coordinates (x1, ..., xn) and consider the invariantization mapping: I : M → Rn, where x1 = Q1 (g) , ..., xn = Qn (g) . At regular points M Σg this mapping I is a covering over a domain Dg ⊂ Rn : I : M Σg → Dg ⊂ Rn.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
The factor equation Emetric is the following equation on a metric h on Rn : Tr Rich = x1, Tr Ric2
h = x2, ..., Tr Ricn h = xn.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
The factor equation Emetric is the following equation on a metric h on Rn : Tr Rich = x1, Tr Ric2
h = x2, ..., Tr Ricn h = xn.
Equivalence classes of metrics ⇐ ⇒ Solutions of PDEs system Emetric.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let M be an oriented 4-dimensional manifold, g be a Lorentzian metric on M.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let M be an oriented 4-dimensional manifold, g be a Lorentzian metric on M. The vacuum Einstein equation Gµν = 0 where Gµν = Rµν − R
2 gµν is the Einstein tensor, with Rµν-Ricci
curvature tensor and R the scalar curvature tensor.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let M be an oriented 4-dimensional manifold, g be a Lorentzian metric on M. The vacuum Einstein equation Gµν = 0 where Gµν = Rµν − R
2 gµν is the Einstein tensor, with Rµν-Ricci
curvature tensor and R the scalar curvature tensor. Let E2 ⊂ J2q be a submanifold corresponding to the equation and let Ek = E(k−2)
2
⊂ Jkq be prolongations of the Einstein equation.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Important remarks:
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Important remarks:
1
The Einstein equation is natural, i.e. invariant of the prolonged action
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Important remarks:
1
The Einstein equation is natural, i.e. invariant of the prolonged action
2
Formally integrable.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Important remarks:
1
The Einstein equation is natural, i.e. invariant of the prolonged action
2
Formally integrable.
3
Fibres of the projections πk : Ek → M are irreducible algebraic manifolds and actions of the differential groups on these fibres are algebraic.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Important remarks:
1
The Einstein equation is natural, i.e. invariant of the prolonged action
2
Formally integrable.
3
Fibres of the projections πk : Ek → M are irreducible algebraic manifolds and actions of the differential groups on these fibres are algebraic. A fibrewise rational function on the bundle πk : Ek → M we call relativistic invariant of order ≤ k if the function is invariant with respect to the prolonged action of the diffeomorphism group.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
The Hodge operator ∗ : Λ2T∗ → Λ2T∗ defines the comlex structure in the bundle of differential 2-forms.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
The Hodge operator ∗ : Λ2T∗ → Λ2T∗ defines the comlex structure in the bundle of differential 2-forms. The Weyl tensor, considered as operator, defines Weyl operator Wg : Λ2T∗ → Λ2T∗, which
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
The Hodge operator ∗ : Λ2T∗ → Λ2T∗ defines the comlex structure in the bundle of differential 2-forms. The Weyl tensor, considered as operator, defines Weyl operator Wg : Λ2T∗ → Λ2T∗, which
commutes with ∗, i.e.Wg is a C-linear operator.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
The Hodge operator ∗ : Λ2T∗ → Λ2T∗ defines the comlex structure in the bundle of differential 2-forms. The Weyl tensor, considered as operator, defines Weyl operator Wg : Λ2T∗ → Λ2T∗, which
commutes with ∗, i.e.Wg is a C-linear operator. TrC Wg = 0.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
The Hodge operator ∗ : Λ2T∗ → Λ2T∗ defines the comlex structure in the bundle of differential 2-forms. The Weyl tensor, considered as operator, defines Weyl operator Wg : Λ2T∗ → Λ2T∗, which
commutes with ∗, i.e.Wg is a C-linear operator. TrC Wg = 0.
Let TrC W2
g
= I1,g + √ −1I2,g, TrC W3
g
= I3,g + √ −1I4,g.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by I1, I2, I3, I4 functions on E2 corresponding to Ii,g.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by I1, I2, I3, I4 functions on E2 corresponding to Ii,g.
They are rational and invariant under diffeomorphisms
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by I1, I2, I3, I4 functions on E2 corresponding to Ii,g.
They are rational and invariant under diffeomorphisms dim(Weyl tensors)-dim (so (1, 3)) = 10 − 6 = 4.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by I1, I2, I3, I4 functions on E2 corresponding to Ii,g.
They are rational and invariant under diffeomorphisms dim(Weyl tensors)-dim (so (1, 3)) = 10 − 6 = 4. Rosenlicht theorem = ⇒ I1, I2, I3, I4 generate relativistic invariants
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by I1, I2, I3, I4 functions on E2 corresponding to Ii,g.
They are rational and invariant under diffeomorphisms dim(Weyl tensors)-dim (so (1, 3)) = 10 − 6 = 4. Rosenlicht theorem = ⇒ I1, I2, I3, I4 generate relativistic invariants
We say that a 3-jet θ3 ∈ E3 is singular (resp. regular) if
dI4 = 0 (resp. dI1 ∧ ... ∧ dI4 = 0) at the point.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by I1, I2, I3, I4 functions on E2 corresponding to Ii,g.
They are rational and invariant under diffeomorphisms dim(Weyl tensors)-dim (so (1, 3)) = 10 − 6 = 4. Rosenlicht theorem = ⇒ I1, I2, I3, I4 generate relativistic invariants
We say that a 3-jet θ3 ∈ E3 is singular (resp. regular) if
dI4 = 0 (resp. dI1 ∧ ... ∧ dI4 = 0) at the point. Denote ∑3 ⊂ E3 the set of all singular 3-jets, it is an algebraic variety.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by I1, I2, I3, I4 functions on E2 corresponding to Ii,g.
They are rational and invariant under diffeomorphisms dim(Weyl tensors)-dim (so (1, 3)) = 10 − 6 = 4. Rosenlicht theorem = ⇒ I1, I2, I3, I4 generate relativistic invariants
We say that a 3-jet θ3 ∈ E3 is singular (resp. regular) if
dI4 = 0 (resp. dI1 ∧ ... ∧ dI4 = 0) at the point. Denote ∑3 ⊂ E3 the set of all singular 3-jets, it is an algebraic variety. At the regular points we write down g = ∑
i≤j
Gij dIi · dIj.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Denote by I1, I2, I3, I4 functions on E2 corresponding to Ii,g.
They are rational and invariant under diffeomorphisms dim(Weyl tensors)-dim (so (1, 3)) = 10 − 6 = 4. Rosenlicht theorem = ⇒ I1, I2, I3, I4 generate relativistic invariants
We say that a 3-jet θ3 ∈ E3 is singular (resp. regular) if
dI4 = 0 (resp. dI1 ∧ ... ∧ dI4 = 0) at the point. Denote ∑3 ⊂ E3 the set of all singular 3-jets, it is an algebraic variety. At the regular points we write down g = ∑
i≤j
Gij dIi · dIj. Then Gij are relativistic invariants of order 3.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Theorem
The field of relativistic invariants is generated by basic invariants (I1, I2, I3, I4) of order 2 and invariants Gij of order 3 and the Tresse derivatives. This field separates regular orbits. The Hilbert function of the field of relativistic invariants is HEin (k) = 2 (k + 3) (k − 1) ,for k ≥ 3.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Theorem
The field of relativistic invariants is generated by basic invariants (I1, I2, I3, I4) of order 2 and invariants Gij of order 3 and the Tresse derivatives. This field separates regular orbits. The Hilbert function of the field of relativistic invariants is HEin (k) = 2 (k + 3) (k − 1) ,for k ≥ 3. The Poincare function is PEin (t) =
2t2(3t3−9t2+6t+2) (1−t)3
.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Theorem
The field of relativistic invariants is generated by basic invariants (I1, I2, I3, I4) of order 2 and invariants Gij of order 3 and the Tresse derivatives. This field separates regular orbits. The Hilbert function of the field of relativistic invariants is HEin (k) = 2 (k + 3) (k − 1) ,for k ≥ 3. The Poincare function is PEin (t) =
2t2(3t3−9t2+6t+2) (1−t)3
. The table of numbers of independent relativistic invariants of pure
k 2 3 4 5 6 7 8 9 HEin 4 24 42 64 90 120 154 192
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let (M, g) be a fixed Einstein manifold, and let Σg ⊂ M the set of singular points of g, (i.e. j3 (g) ∈ Σ3 ). Assume that Σg = M.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let (M, g) be a fixed Einstein manifold, and let Σg ⊂ M the set of singular points of g, (i.e. j3 (g) ∈ Σ3 ). Assume that Σg = M. Take space C2 with fixed coordinates (z1, z2) and consider the invariantization mapping: I : M → C2, where z1 = I1,g + √ −1I2,g, z2 = I3,g + √ −1I4,g.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let (M, g) be a fixed Einstein manifold, and let Σg ⊂ M the set of singular points of g, (i.e. j3 (g) ∈ Σ3 ). Assume that Σg = M. Take space C2 with fixed coordinates (z1, z2) and consider the invariantization mapping: I : M → C2, where z1 = I1,g + √ −1I2,g, z2 = I3,g + √ −1I4,g. At regular points M Σg this mapping is a covering over a domain Dg ⊂ C2 : I : M Σg → Dg ⊂ C2.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
The factor equation EEin is the following equation on a metric h on C2 : Gh = 0, TrC W2
h = z1,
TrC W3
h = z2.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
The factor equation EEin is the following equation on a metric h on C2 : Gh = 0, TrC W2
h = z1,
TrC W3
h = z2.
Equivalence classes of Einstein metrics ⇐ ⇒ Solutions of system EEin.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
The factor equation EEin is the following equation on a metric h on C2 : Gh = 0, TrC W2
h = z1,
TrC W3
h = z2.
Equivalence classes of Einstein metrics ⇐ ⇒ Solutions of system EEin. Remark on space-time problem.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let M be an oriented 4-dimensional manifold, g be a Lorentzian metric and F be a differential 2-form on M.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let M be an oriented 4-dimensional manifold, g be a Lorentzian metric and F be a differential 2-form on M. The Einstein-Maxwell equation: G = T , dF = 0, d ∗ F = 0, where G is the Einstein tensor, F is the Faraday tensor and ∗ is the Hodge operator.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let M be an oriented 4-dimensional manifold, g be a Lorentzian metric and F be a differential 2-form on M. The Einstein-Maxwell equation: G = T , dF = 0, d ∗ F = 0, where G is the Einstein tensor, F is the Faraday tensor and ∗ is the Hodge operator. T is the energy-momentum tensor: Tij = 2k c4
j + 1
4gijFklF kl
where k gravitational constant and c is the velocity of light.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let τ(0,2) = S2T∗ ⊕ Λ2T∗ = T∗ ⊗ T∗ be the bundle of (0, 2)-tensors and let EM2 ⊂ J2τ(0,2) be the submanifold corresponding to Einstein-Maxwell equation.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let τ(0,2) = S2T∗ ⊕ Λ2T∗ = T∗ ⊗ T∗ be the bundle of (0, 2)-tensors and let EM2 ⊂ J2τ(0,2) be the submanifold corresponding to Einstein-Maxwell equation. Denote by EMk = EM(k−2)
2
⊂ Jkτ(0,2) their prolongations.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Note that Einstein-Maxwell equation is
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Note that Einstein-Maxwell equation is
natural
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Note that Einstein-Maxwell equation is
natural formally integrable
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Note that Einstein-Maxwell equation is
natural formally integrable Fibres of the projections πk : EMk → M are irreducible algebraic manifolds and actions of the differential groups
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Note that Einstein-Maxwell equation is
natural formally integrable Fibres of the projections πk : EMk → M are irreducible algebraic manifolds and actions of the differential groups
A fibrewise rational function on the bundle πk : EMk → M we call electromagnetic invariant of order ≤ k if the function is invariant with respect to the prolonged action of the diffeomorphism group.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let s = (g, F) ∈ τ(0,2). and let Ωg be the corresponding g unit volume form on M.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let s = (g, F) ∈ τ(0,2). and let Ωg be the corresponding g unit volume form on M. Then relations F ∧ F = I1 (s) Ωg, F ∧ ∗F = I2 (s) Ωg, define two electromagnetic invariants I1, I2 of order 0.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let s = (g, F) ∈ τ(0,2). and let Ωg be the corresponding g unit volume form on M. Then relations F ∧ F = I1 (s) Ωg, F ∧ ∗F = I2 (s) Ωg, define two electromagnetic invariants I1, I2 of order 0. We say that point (ga, Fa) ∈ T(0,2)
a
, a ∈ M, is Rainich regular if (I1, I2) = 0 at this point.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let s = (g, F) ∈ τ(0,2). and let Ωg be the corresponding g unit volume form on M. Then relations F ∧ F = I1 (s) Ωg, F ∧ ∗F = I2 (s) Ωg, define two electromagnetic invariants I1, I2 of order 0. We say that point (ga, Fa) ∈ T(0,2)
a
, a ∈ M, is Rainich regular if (I1, I2) = 0 at this point. At Rainich regular points one has splitting Ta = Ha ⊕ Ea
elliptic planes.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
We say that a point in J1τ(0,2) is Rainich regular if its projection in τ(0,2) is regular and restrictions of the total differentials dI1 and dI2
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
We say that a point in J1τ(0,2) is Rainich regular if its projection in τ(0,2) is regular and restrictions of the total differentials dI1 and dI2
Denote by e∗
1, e∗ 2, e∗ 3, e∗ 4 the corresponding coframe.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
We say that a point in J1τ(0,2) is Rainich regular if its projection in τ(0,2) is regular and restrictions of the total differentials dI1 and dI2
Denote by e∗
1, e∗ 2, e∗ 3, e∗ 4 the corresponding coframe.
Remark that the dual frame defines invariant total derivations ∇1, ∇2, ∇3, ∇4 of order 1.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
We say that a point in J1τ(0,2) is Rainich regular if its projection in τ(0,2) is regular and restrictions of the total differentials dI1 and dI2
Denote by e∗
1, e∗ 2, e∗ 3, e∗ 4 the corresponding coframe.
Remark that the dual frame defines invariant total derivations ∇1, ∇2, ∇3, ∇4 of order 1. Let Gij and Fij the components of tensors g and F in this coframe. They are electromagnetic invariants of order 1.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Theorem
The field of relativistic electromagnetic invariants is generated by invariants: I1, I2 of order 0, invariants Gij , Fij of order 1 and invariant derivations ∇1, ∇2, ∇3, ∇4. This field separates Rainich regular orbits. The Hilbert function of the field of electromagnetic invariants is HEM (0) = 2, HEM (1) = 14, HEM (k) = 4k (k + 3) , for k ≥ 2.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Theorem
The field of relativistic electromagnetic invariants is generated by invariants: I1, I2 of order 0, invariants Gij , Fij of order 1 and invariant derivations ∇1, ∇2, ∇3, ∇4. This field separates Rainich regular orbits. The Hilbert function of the field of electromagnetic invariants is HEM (0) = 2, HEM (1) = 14, HEM (k) = 4k (k + 3) , for k ≥ 2. The Poincare function is PEM (t) = 2
.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let J = {J1, J2., J3, J4 } be for electromagnetic invariants. By regular jet we undersand jets where
dJ4 = 0.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let J = {J1, J2., J3, J4 } be for electromagnetic invariants. By regular jet we undersand jets where
dJ4 = 0. Denote by ΣJ ⊂ Jkτ(0,2) the set of J-singular jets.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let J = {J1, J2., J3, J4 } be for electromagnetic invariants. By regular jet we undersand jets where
dJ4 = 0. Denote by ΣJ ⊂ Jkτ(0,2) the set of J-singular jets. Let (M, g, F) be a solution ofEinstein- Maxwell and let Σg,F ⊂ M be the set of J-singular points of s = (g, F). Assume that Σg,F = M.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Let J = {J1, J2., J3, J4 } be for electromagnetic invariants. By regular jet we undersand jets where
dJ4 = 0. Denote by ΣJ ⊂ Jkτ(0,2) the set of J-singular jets. Let (M, g, F) be a solution ofEinstein- Maxwell and let Σg,F ⊂ M be the set of J-singular points of s = (g, F). Assume that Σg,F = M. Take space R4 with fixed coordinates (x1, x2, x3, x4) and consider the invariantization mapping: I : M → R4, where x1 = J1 (g, F) , x2 = J2 (g, F) , x3 = J3 (g, F) , x4 = J4 (g, F) .
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Define as above invariants Gij and Fij g = ∑
i,j
Gij dIi · dIj and F = ∑
i,j
Fij dIi ∧ dIj.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Define as above invariants Gij and Fij g = ∑
i,j
Gij dIi · dIj and F = ∑
i,j
Fij dIi ∧ dIj. At regular points M Σg,F this mapping is a covering over a domain Dg,F ⊂ R4 : I : M Σg,F → Dg,F ⊂ R4.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Define as above invariants Gij and Fij g = ∑
i,j
Gij dIi · dIj and F = ∑
i,j
Fij dIi ∧ dIj. At regular points M Σg,F this mapping is a covering over a domain Dg,F ⊂ R4 : I : M Σg,F → Dg,F ⊂ R4. The factor equation EEM is the following equation on a metric h and differenatial 2-form F on R4 : Gh = Th,F , dF = 0, d ∗ F = 0, J1 (g, F) = x1, J2 (g, F) = x2, J3 (g, F) = x3, J4 (g, F) = x4.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Define as above invariants Gij and Fij g = ∑
i,j
Gij dIi · dIj and F = ∑
i,j
Fij dIi ∧ dIj. At regular points M Σg,F this mapping is a covering over a domain Dg,F ⊂ R4 : I : M Σg,F → Dg,F ⊂ R4. The factor equation EEM is the following equation on a metric h and differenatial 2-form F on R4 : Gh = Th,F , dF = 0, d ∗ F = 0, J1 (g, F) = x1, J2 (g, F) = x2, J3 (g, F) = x3, J4 (g, F) = x4. Equivalence classes of Einstein - Maxwell solutions ⇐ ⇒ Solutions of PDEs
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Define as above invariants Gij and Fij g = ∑
i,j
Gij dIi · dIj and F = ∑
i,j
Fij dIi ∧ dIj. At regular points M Σg,F this mapping is a covering over a domain Dg,F ⊂ R4 : I : M Σg,F → Dg,F ⊂ R4. The factor equation EEM is the following equation on a metric h and differenatial 2-form F on R4 : Gh = Th,F , dF = 0, d ∗ F = 0, J1 (g, F) = x1, J2 (g, F) = x2, J3 (g, F) = x3, J4 (g, F) = x4. Equivalence classes of Einstein - Maxwell solutions ⇐ ⇒ Solutions of PDEs Remark on space-time problem.
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
The tables of numbers of independent metric, relativistic and electromagnetic invariants of pure order k for small values k : k 1 2 3 4 5 6 7 Metric 14 60 126 224 360 540 Einstein 4 24 42 64 90 120 Einstein-Maxwell 2 14 40 72 112 160 216 280
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39
Lychagin & Yumaguzhin (University of Tromso) Differential Invariants Workshop on “Infinite-dimensional Riemannia / 39