An existence theorem for $G$-structure preserving affine immersions

Abstract

We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of isometric immersions into products of space forms.

Associated Family of G-Structure Preserving Minimal Immersions in Semi-Riemannian Manifolds

“…Throughout this article we use the concepts of Christoffel tensor and inner-torsion of an affine manifold with G-structure (see [10]). To define the Christoffel tensor we observe that, if π : E → M is a vector bundle with typical fiber E 0 and s : U → FR E0 (E) is a smooth local E 0 -frame of E, we can define a connection d s on E| U by: Let now π : E → M be a vector bundle with typical fiber E 0 endowed with a connection ∇, let G be a Lie subgroup of GL(E 0 ) and P ⊂ FR E0 (E) be a G-structure on E. For each x ∈ M , denote by G x ⊂ GL(E x ) the Lie subgroup consisting of all G-structure preserving isomorphisms of E x , i.e., g ∈ G x if and only if g • p ∈ P x for all p ∈ P x ; denote by g x ⊂ gl(E x ) its Lie algebra.…”

“…It's not hard to see that ω is in fact the Christoffel tensor Γ with respect to s. Define I P x : T x M → gl(E x )/g x as the composition of maps illustrated in the following diagram: We call I P x the inner torsion of the G-structure P at the point x with respect to the connection ∇. It can be proved I P x does not depend on the choice of the local section s (see [10,Section 5]) and that I P x : T x M → gl(E x )/g x is given by the composition of the Christoffel tensor Γ x : T x M → gl(E x ) of the connection ∇ with respect to s and the quotient map gl(E x ) → gl(E x )/g x (see [10]). …”

“…The following proposition, from [10], gives us some information about the uniqueness of solutions of the semi-Riemannian isometric immersion problem. Proposition 6.4.…”

“…Consider the component α F ∈ Γ(Lin(E; F ; F ⊥ )) of ∇ with respect to the decomposition E = F ⊕ F ⊥ , this is the second fundamental form with respect to the decomposition E = F ⊕ F ⊥ . If we look at ∇ v g : E x × E x → R as a linear endomorphism of E x , we have: [10,Example 5.4]). …”

“…In Sect. 3 we will give a slightly modified statement of an isometric immersion theorem proved in [10], more suited for the purposes of the present paper. Section 4 gives a very short account of the basic facts about minimal immersions.…”