I am a bigginer in differential geometry, i am confused about the following two notions:
Let $(S_1,d_1)$ and $(S_2,d_2)$ be two surfaces (endowed with metrics $d_1,d_2$)
Can we talk about isometry between $S_1$ and $S_2$ without talking about metrics ? (because sometimes i find the sentence "two isometric surfaces"!! but they don t talk about metrics?? "
If $(S_1,d_1)$ and $(S_2,d_2)$ are "diffeomorphic" do this imply that they are isometric or the reverse is true ? What is stronger the notion "diffeomorphic" or "isometric"
If the surfaces are said to be isometric but the metrics are not explicitly mentioned, then they must be known from context. For example, if these are given in terms of embeddings in Euclidean space, the standard metric of that Euclidean space is implied.
Diffeomorphic does not imply isometric. Examples are easy to construct.