On Isometric Immersion of Three-Dimensional Geometries $SL_2$, $Nil$ and $Sol$ into a Four-Dimensional Space of Constant Curvature

Abstract

We prove the nonexistence of isometric immersion of geometries $\text{Nil}^3$, $\widetilde{SL}_2$ into the four-dimensional space $M_c^4$ of the constant curvature $c$. We establish that the geometry $\text{Sol}^3$ cannot be immersed into $M_c^4$ if $c \neq -1$ and find the analytic immersion of this geometry into the hyperbolic space $H^4(-1)$