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The title pretty much sums it up. I am wondering if anyone has some interesting research topics to investigate about non-euclidian geometry?

Sorry for the confusion, I am a first year undergrad, so something like riemannian geometry with like manifold seems out of my math capabilities. I know this area is already "dead", I just need to write a 4000 paper about it.

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    $\begingroup$ This question seems incredibly broad. Basically all Riemannian geometry is about non-Euclidean geometry, it is a whole field with hundreds of active researchers and hundreds of questions. Maybe you could say a little more about what you seek, for what purpose, to make the question answerable, but in any case it might still be deemed unsuitable for MO. $\endgroup$
    – Benoît Kloeckner
    Commented Nov 13, 2021 at 8:42
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    $\begingroup$ @BenoîtKloeckner If geometry is not euclidean, it does not mean that it is non-euclidean. For example, Riemannian geometry is not considered to be non-euclidian. $\endgroup$
    – Anton Petrunin
    Commented Nov 13, 2021 at 18:12
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    $\begingroup$ You made wrong tags to the question, so it is misunderstood by many. $\endgroup$
    – Anton Petrunin
    Commented Nov 13, 2021 at 18:19
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    $\begingroup$ @AntonPetrunin then terminology differs between people: I do consider Riemannian geometry as non-Euclidean. Do you restrict the word to constant curvature? To homogeneous spaces? In any case, the question cannot really be answered as is. $\endgroup$
    – Benoît Kloeckner
    Commented Nov 13, 2021 at 22:49
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    $\begingroup$ @BenoîtKloeckner in the standard terminology is non-euclidean geometry is a geometry that like euclidean, but not euclidean. $\endgroup$
    – Anton Petrunin
    Commented Nov 14, 2021 at 4:34

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