An embedded surface is a surface placed in an ambient space without self-intersections. A surface 
is 
-embeddable if it can be placed in 
-space without self-intersections, but cannot be similarly
placed in any 
for 
.
The Costa minimal surface and gyroid
are embeddable in 
,
but the Klein bottle is not (the commonly depicted
representation requires the surface
to pass through itself).
The study of complete minimal surfaces that are also embedded is an important part of the theory of minimal surfaces.
." Ann. Math. 145, 1-31, 1997.Hoffman,
D. and Karcher, H. "Complete Embedded Minimal Surfaces of Finite Total Curvature."
In