Injective Patch

An injective patch is a Patch such that ${\bf x}(u_1,v_1)={\bf x}(u_2,v_2)$ implies that $u_1=u_2$ and $v_1=v_2$. An example of a Patch which is injective but not Regular is the function defined by $(u^3, v^3, uv)$ for $u,v \in(-1,1)$. However, if ${\bf x}:U\to\Bbb{R}^n$ is an injective regular patch, then x maps $U$ diffeomorphically onto ${\bf x}(U)$.

See also Patch, Regular Patch

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 187, 1993.