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We study some geometric properties associated with the $t$-geometric means $A{♯}_{t}B:=A^{1/2}{\left(A^{-1/2}B{A}^{-1/2}\right)}^t{A}^{1/2}$ of two $n\times n$ positive definite matrices $A$ and $B$. Some geodesical convexity results with respect to the Riemannian structure of the $n\times n$ positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding $m$ pairs...