How can I show that two isometries $F_1,F_2 : M \to M$ which agree at point $p$ and induce the same linear mapping from $T_pM$ agree on a neighborhood on $p$?
I am happy about a hint or an idea of how to approach it would be much appreciated.
2026-02-22 16:03:12.1771776192
Show that two isometries induce the same linear mapping
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Hint: Use that an isometry sends geodesics to geodesics, and check what happens to geodesics emanating from $p$.