Hi Dears,
Just read the paper I have shared below and answer the question.
https://www.ams.org/journals/proc/2013-141-03/S0002-9939-2012-11743-5/S0002-9939-2012-11743-5.pdf
I have shared the pdf version of the paper with you all below (It is open access).
On page 1032 the authors introduced (S) property and have said that: "It is not hard to check that symmetric Hadamard Manifolds satisfy in this property (exp_x^{-1} y acts in a role of [xy] in Hadamard spaces)". Now the question is how?
If someone could answer Kakavandi's question, I will accept having collaboration in this issue with her/him.
Regards
Farshid Khojasteh