For every $\epsilon,R>0$ there exists $\delta=\delta(n,\epsilon,R)>0$ such that if ${\rm Ric}_{M^{n}}\geq-(n-1)\delta$ then:

1. (1)

   If $u:B_{2R}(p)\to\mathds{R}^{k}$ is a $\delta$-splitting map, then there exists a map $f:B_{R}(p)\to u^{-1}(0)$ such that

   $$(u,f):B_{R}(p)\to\mathds{R}^{k}\times u^{-1}(0)\,,$$

   is an $\epsilon$-Gromov Hausdorff map, where $u^{-1}(0)$ is given the induced metric.

2. (2)

   If

   $\displaystyle d_{GH}(B_{\delta^{-1}}(p),B_{\delta^{-1}}(0))<\delta,$

   (1.13)

   where $0\in\mathds{R}^{k}\times Y$, then there exists an $\epsilon$-splitting map $u:B_{R}(p)\to\mathds{R}^{k}$.