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Existence of conformal metrics with constant Gauss curvature
In the well-known problem studied by Poincaré, we are interested in finding out whether or not a Riemannian surface is conformally equivalent to a constant Gauss curvature Riemannian surface. The problem is generalized in higher dimensions by replacing the Gauss curvature with the scalar curvature. This is known as the Yamabe problem. Let (M, g0) be a Riemannian surface. As shown previously, one can consider an orthonormal moving coframe on which the metric is expressed locally as: g0 = η1 ⊗ η1 + η2 ⊗ η2. For an arbitrary metric g0, the Gauss curvature has a higher chance to be non-constant. We also know that multiplying the metric by a positive number R2 changes the Gauss curvature form K to K/R2, and hence, a non-constant Gauss curvature remains non-constant by just dilating the metric, however, one can imagine that multiplying the metric by a non-vanishing function can neutralize the variation of the Gauss curvature. So, is it possible to find a metric g, conformally equivalent to g0, such that (M, g) is of constant Gauss curvature? Namely, is it possible to choose a function λ such that the Riemannian surface (M, g), where g = e2g0, is of constant Gauss curvature? Proposition 1.23 Prescribed Gauss curvature Let (M, g0) be a Riemannian surface.
Cartan–Kähler theory
If an EDS contains differential 1-forms and functions, we can still apply the Frobenius theorem to the submanifold defined by the vanishing of these functions (except on the possible singularities). However, if the EDS contains differential forms of a degree greater than 1, the Frobenius theorem is no longer helpful, as is often the case for EDSs arising from geometric problems. For instance, let be a closed differential 2-form on an 2m-dimensional manifold M such that m 6= 0. The pair (M2m, ) is called a symplectic manifold. The integral mdimensional manifolds of { }, if they exist, are called Lagrangian manifolds. Thus finding Lagrangian manifolds for a given symplectic manifold is equivalent to looking for integral manifolds of a differential 2-form. Besides the Frobenius theorem, there are standard differential techniques of ordinary differential equations that allow a complete (local) description of integral manifolds of a exterior differential system, like the Pfaff-Darboux and Goursat theorems. The following theory represents a general method for finding and constructing integral manifolds for any exterior differential system.
Integral elements of an EDS and their extensions
Definition 2.18 Integral element Let I be an exterior differential ideal on an mdimensional manifoldM. An integral element of I at a point M ∈Mis a linear subspace E of TMM such that ϕE = 0 for all ϕ ∈ I, where ϕE means the evaluation of ϕ on any basis of E. The set of p-dimensional integral elements of I is denoted by Vp(I).
Proposition 2.19 Subspace of an integral element If E is an integral element of the exterior differential ideal I on M, then every vector subspace of E is also an integral element of I.
Proof. Let W1 be a vector subspace of an n-integral element E of I such that W1 is not an integral element of I. Then, there exists a differential form ϕ ∈ I, such that ϕW1 6= 0. Let W2 be a vector subspace of E such that E = W1 ⊕W2. Then ϕ ∧ ψ, where we choose ψ such that ψW1 = 0 and ψW2 6= 0, and the degree of ϕ is dimW1, belongs to I and does not annihilate E, contradicting the assumption that E is an integral element of I.
Proposition 2.20 Integral elements space of an EDI Let I be an exterior differential ideal on an m-dimensional manifold. Then Vp(I) = {E ∈ Gp(TM)|ϕE = 0 for all ϕ ∈ Ip}.
Integral flags, involution and existence theorems
In this subsection, we are interested in determining whether or not an exterior differential system which has a condition of independence admits integral manifolds. Such condition is present, for instance, for exterior differential systems arising from systems of PDEs. It is then compulsory that the defining equations of the integral manifold do not contain relations between the independent variables.
Definition 2.27 EDS with an independence condition An EDS in an m-dimensional manifold M with independence condition is a pair (I, ) where I is an exterior differential system on M and is a differential non-vanishing n-form on M. Exterior ideal and exterior differential ideal with an independence condition are defined like an EDS, i.e., by the assignment of a non-vanishing differential n-form.
Definition 2.28 Integral elements with an independence condition Let I be an exterior differential ideal on an m-dimensional manifold M with an independence condition ∈ (∧nT∗M), and let Gn(TM, ) = {E ∈ Gn(TM)/E 6= 0} be the Grassmannian manifold of the tangent bundle TM, consisting of the n-dimensional subspaces TM on which does not vanish. Then the set of integral elements of (I, ) denoted by Vn(I, ) is the set of integral elements of I on which does not vanish, i.e., Vn(I, ) = Vn(I) ∩ Gn(TM, ). Consequently, solutions to (I, ) are integral manifolds of I on which does not vanish.
Definition 2.29 Kähler ordinary integral element An n-integral element E of an exterior differential ideal is said to be Kähler ordinary if there exists a differential n-form such that E 6= 0 with with the property that E is an ordinary zero of the set of functions F = {ϕ|ϕ ∈ In}.
Table of contents :
Introduction en français
Introduction in English
1 Cartan’s structure equations
1.1 Connection on a vector bundle
1.2 The tangent bundle case
1.3 Applications to surfaces
1.3.1 Christoffel symbols and Gauss curvature
1.3.2 Existence of conformal metrics with constant Gauss curvature
2 EDS and Cartan–Kähler theory
2.1 Exterior differential systems
2.2 Cartan–Kähler theory
2.2.1 Integral elements of an EDS and their extensions
2.2.2 Integral flags, involution and existence theorems
3 Some surface embedding results
3.1 Lagrangian surfaces
3.2 Isometric embedding of surfaces
3.3 Isometric Lagrangian embedding of surfaces
3.A Lagrangian manifolds in R2m
3.B The Cartan–Janet theorem
4 On generalized isometric embeddings
4.1 Conervation laws
4.2 The generalized isometric embedding problem
4.3 Motivations
4.3.1 The isometric embedding problem
4.3.2 Harmonic maps between Riemannian manifolds
4.4 On generalized isometric embedding results
4.5 Application to energy-momentum tensors
4.A Detailed proof of lemma 4.16 for surfaces
5 A general strategy and the conservation law case
5.1 The generalized isometric embedding problem via EDS
5.2 Specialization in the conservation law case
5.2.1 Proof of lemma 5.14
5.2.2 Another proof of theorem 4.12
5.A The case (∧2TM3, g,M3,∇, Id∧2TM3)
6 Other generalized isometric embedding results
6.1 Covariantly closed differential 1-forms
6.2 Generalized isometric embedding of 2-form with anti-self dual condition
6.A Remarks on generalized Bianchi identities
6.A.1 (V2,M3, g,∇, φ)1
6.A.2 (V3,M3, g,∇, φ)1
Appendix
A Computations and proofs
B Tableaux and linear Pfaffian systems
B.1 A short review of tableaux and linear Pfaffian systems
B.2 Applications
B.2.1 Heat equation
B.2.2 Conformal embeddings
B.2.3 Lagrangian manifolds in Cm
Bibliography
