- #1
demonelite123
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proving that the sphere cannot be expressed in "flat coordinates"
for the sphere in R^3 i have that ds^2 = dϕ^2 + sin^2(ϕ)dθ^2. using the definition of a flat space as one where given a set of curvilinear coordinates, one can find a metric such that ds^2 = dx^2 + dy^2, how would one prove that the sphere is not a flat space which means it cannot have "flat coordinates"?
i tried setting dϕ^2 + sin^2(ϕ)dθ^2 = dx^2 + dy^2 to find a contradiction. i was thinking of taking two points and finding the difference between them to approximate the differentials. however i have no idea what to do with the dx and dy on the right side as the coordinate system i am in do not involve them at all. any help is greatly appreciated!
for the sphere in R^3 i have that ds^2 = dϕ^2 + sin^2(ϕ)dθ^2. using the definition of a flat space as one where given a set of curvilinear coordinates, one can find a metric such that ds^2 = dx^2 + dy^2, how would one prove that the sphere is not a flat space which means it cannot have "flat coordinates"?
i tried setting dϕ^2 + sin^2(ϕ)dθ^2 = dx^2 + dy^2 to find a contradiction. i was thinking of taking two points and finding the difference between them to approximate the differentials. however i have no idea what to do with the dx and dy on the right side as the coordinate system i am in do not involve them at all. any help is greatly appreciated!