Given a Riemann manifold $(M,g)$,the Riemann metric induces a topology on $M$ which given by $d(p,q)$=the shortest length between $p$ and $q$,it's a metric topology,and my question is:is this topology the topology of $M$ as a manifold?If not,then what is the relation between them?
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@C weid,you can see it as a proposition in the lee's book(introduction to smooth manifolds) in the section that is about riemanniean metric,you can find riemannian metric in the contents. – R Salimi Dec 19 '13 at 11:00
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@RSalimiThank you I'm just reading it.Excellent book,rather suitable for beginners like me... – C Weid Dec 19 '13 at 11:10
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See answer here : http://math.stackexchange.com/questions/612754/is-the-usual-topology-on-the-upper-half-plane-same-as-that-induced-by-riemannian. – Seub Dec 19 '13 at 21:37