Geodesic
A Geometric Mean for Symmetric Spaces of Noncompact Type
Journal of Lie theory, Tome 24 (2014) no. 3, pp. 725-736.

Voir la notice de l'article provenant de la source Heldermann Verlag

The concept of the t-geometric mean of two positive definite matrices is extended to symmetric spaces of noncompact type. The t-geometric mean of two points in such a symmetric space yields the unique geodesic joining the points and the geometric mean is the midpoint. A parametrization of the geodesic in terms of the two points is given. Inequalities about geometric mean and geodesic triangle are given in terms of Kostant's pre-order on semisimple Lie groups as well as on their Lie algebras.
Classification : 15A45, 15A48, 53C35
Mots-clés : Geometric mean, positive definite matrices, symmetric spaces, semisimple Lie groups, geodesics, log majorization, Kostant's order 
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     author = {M. Liao and X. Liu and T.-Y. Tam },
     title = {A {Geometric} {Mean} for {Symmetric} {Spaces} of {Noncompact} {Type}},
     journal = {Journal of Lie theory},
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M. Liao; X. Liu; T.-Y. Tam . A Geometric Mean for Symmetric Spaces of Noncompact Type. Journal of Lie theory, Tome 24 (2014) no. 3, pp. 725-736. http://geodesic.mathdoc.fr/item/JLT_2014_24_3_JLT_2014_24_3_a6/