Isometric immersions of space forms into Riemannian and pseudo-Riemannian spaces of constant curvature

Abstract

This survey contains results on local and global isometric immersions of two-dimensional and multidimensional Riemannian and pseudo-Riemannian space forms into spaces of constant sectional curvature.

Isometric immersions of $${{\mathbb R}^2}$$ into $${{\mathbb R}^4}$$ and perturbation of Hopf tori

“…R ⊥ ≡ 0. This geometric setting contains a large number of examples, and has been widely studied (see [1][2][3][6][7][8][9][10]15,17,18,[23][24][25][27][28][29]). In this work we give a global description of all simply connected flat surfaces with flat normal bundle in R 4 , and we use it as main application to study flat tori in R 4 with R ⊥ ≡ 0 and regular Gauss map.…”

Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems

“…(See [5] for an account of related regularity results.) Let X be a closed simply connected Riemannian C an -manifold of dimension n ≥ 1, and let f ∶ X → R q be an isometric C ∞ -immersion.…”

“…(The former is an analytic problem lying beyond our present means, while the latter, probably, may be resolved by purely algebraic means; see section 3.6.2.) 5 A rough counting-parameters argument shows that C r for r > (s n − 1)n + 1 is smooth enough for this purpose, where a better bound on r follows from the Gauss formula. 6 The existence of smooth isometric immersions of (compact?)…”

“…It is (almost) obvious that this map is C 1,α , for all α < 1, probabaly even C 65 Certain obstructions for q < 2n, some of which are applicable to compact flat n-manifolds minus m ≤ n small balls are presented in 3.2.1 in [35] 1986; also see [5]. 66 The best result of this kind for n = 2 is Burago's isoperimetric inequality; see [12] and 3.2.3 in [35].…”

On Invariants of Immersions of an n-Dimensional Manifold in an n-Dimensional Pseudo-Euclidean Space*

“…Isometric immersions of pseudoRiemannian space forms have been a subject of wide interest (see [4,6,7]). …”