And I just want to add as a physicist: The most interesting manifolds are the differentiable ones, because there you can do general relativity! But manifolds are also relevant in other, more unexpected places, like a pendulum: It moves on a submanifold of R^3 due to the constraint of the string.
And I just want to add as a physicist: The most interesting manifolds are the differentiable ones, because there you can do general relativity! But manifolds are also relevant in other, more unexpected places, like a pendulum: It moves on a submanifold of R^3 due to the constraint of the string.