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Let $E$ be a Banach space with the $c^1$-norm $\|\cdot\|$ in $ E \backslash\{0\}$ and $S(E)=\{e\in E: \|e\|=1\}.$ In this paper, a geometry characteristicfor $E$ is presented by using a geometrical construct of $S(E).$ That is, thefollowing theorem holds : the norm of $E$ is of $c^1$ in $ E \backslash \{0\}$if and only if $S(E)$ is a $c^1$-submanifold of $E,$ with ${\rm codim}S(E)=1.$The theorem is very clear, however, its proof is non-trivial, which shows anintrinsic connection between the continuous differentiability of the norm$\|\cdot\|$ in $ E \backslash \{0\}$ and differential structure of $S(E).$

A Geometry Characteristic for Banach Space with $c^1$-Norm