opal zealot
- "A differential mannifold is a topological mannifold with a maximal atlas on it"
Why do we care about maximal atlases on mannifolds? In practice we always use an atlas with the least amount of patches and a least patches atlas is in the same class of equivalence with its maximal atlas so we should choose a representative of that class as something simple and practical. I can choose Z_3 = { 12, -8, 300974 } but they aren't good represantives, are they? Or maybe there is a reason for that and I made this post for nothing.
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An atlas is (k-)compatbile if all of its patches are comptabile between themselves (i.e. for any 2 patches in the atlas (U,f) and (V, g) then either (U and V are disjoint) or ( U and V are NOT disjoint and f * g^-1 and g * f^-1 are k differentiable on their respective restricted domains). Does this mean that if we have an atlas which has homeomorphisms as their transition maps f,g,h,... but ONLY miraculously k differentiable on their intersecting domains, is it considered a manifold ( I just want to see if this is really what the definition says )
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If we have a curve on that differential mannifold going from a point x to y (x!=y) and it goes through different patches, how do we describe the coordonates on the differential mannifold? since it's differential it has an atlas and the transition of coordonates is smooth and with no problems but it feels weird to transition for a point with some coordonates and then possible for a closely point on that curve which is in a different patch has different system of coordonates.
Please correct my little understanding of this subject if I am asking something wrong
prisma pumice