Let \(\P:=\P_m\) be the space of all \(m\times m\) positive definite matrices. On the one hand, it is a convex cone of \(\mathbb H:=\mathbb H_m\), the Euclidean space of \(m\times m\) Hermitian matrices. On the other hand, it is a Riemannian-Cartan manifold and a simply connected complete Riemannian manifold with non-positive sectional curvature, if we equip \(\P\) with the Riemannian metric \[ \langle X,Y \rangle := \mathrm{Tr} (A^{-1}XA^{-1}Y),\qquad A\in P, \;X, Y\in \mathbb H. \] Here, the tangent space of \(\P\) at any point \(A\in \P\) is identified with \(\mathbb H\) via the exponential map. The Riemannian distance is \(\delta (A,B):=\|\log A^{-1/2}BA^{-1/2}\|_2\). Given any two points \(A, B\in \P\), the unique geodesic connecting \(A\) to \(B\) has a parametrization \(t\to A\sharp _t B:=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}\). The (two-variable) geometric mean \(A\sharp B\) of \(A, B\) is the midpoint of the geodesic, that is, \(A\sharp B := A\sharp_{1/2}B\), which is the unique \(\delta\)-midpoint between \(A\) and \(B\). The multivariate extension of the two-variable geometric mean is the Cartan mean of \(A_1, \dots , A_n\in \P\), which is the unique minimizer \[ \Lambda_n (A_1, \dots, A_n) := \arg\min_{X\in \P} \sum_{j=1}^n\delta^2(A_j, X). \] It arises also as a unique positive definite solution \(X\) of the Karcher equation (gradient zero equation) \[ \sum^n_{j=1} \log (X^{-1/2}A_jX^{-1/2}) = 0. \] The function \(\Lambda_n:\P^n\to \P\) has the property of repetition invariancy: \[ \Lambda_n (A_1, \dots, A_n) = \Lambda_{nk}({\mathbb A}, \dots , {\mathbb A}), \] where the block \({\mathbb A}:=(A_1, \dots, A_n)\) has \(k\) occurrences on the right side. The authors construct a new geometric mean invariant under repetitions and the key idea is to consider the modified Karcher equation with the weighted geometric mean \(A_i\sharp_t A_j\) instead of \(A_j\): \[ \sum^n_{i, j=1} \log (X^{-1/2} (A_i\sharp_t A_j)X^{-1/2}) = 0,\qquad t\in [0, 1]. \] It has the unique solution \[ \Lambda_{t,n}^\dagger (A_1, \dots, A_n) : = \Lambda_{n^2}(A_1\sharp_t {\mathbb A}, \dots, A_n\sharp_t {\mathbb A}), \] where \(A_j\sharp_t {\mathbb A} = (A_j\sharp_t A_1, \dots, A_j\sharp_t A_n)\in \P^n\). The authors show that \(\Lambda_t^\dagger =\{\Lambda_{t,n}^\dagger\}_{n\geq 1}\) is a geometric mean invariant under repetitions and is in general distinct from the Cartan mean \(\Lambda = \Lambda_0^\dagger = \Lambda_1^{\dagger}\). They also review a geometric mean satisfying the Ando-Li-Mathias axioms (see [\textit{T. Ando} et al., Linear Algebra Appl. 385, 305--334 (2004; Zbl 1063.47013)]), and give a construction of the repetition invariant geometric mean from the given geometric mean. They further consider an extension of repetition invariant geometric means to the contractive barycenters of integrable probability measures, and obtain important properties including invariance under inversion and congruence transformation, monotonicity, and arithmetic-geometric-harmonic mean inequalities. Inequalities from the derived geometric means including the Yamazaki inequality and unitarily invariant norm inequalities are obtained. Some open problems are presented.