Spivak's definition requires that they admit a metric, but this is not really necessary for most purposes. The Nash embedding theorem states that those having metric tensors can be isometrically embedded in Euclidean space. They are typically required to be Hausdorff and second countable, but must always be locally Euclidea
Spivak's definition requires that they admit a metric, but this is not really necessary for most purposes. The Nash embedding theorem states that those having metric tensors can be isometrically embedded in Euclidean space. They are typically required to be Hausdorff and second countable, but must always be locally Euclidean. The differentiable type has smooth transition maps, unlike the topological version, and the Riemannian type has a metric tensor. The Poincar?conjecture states that all simply connected closed examples of these in 3-space are homeomorphic to the 3-sphere. FTP, name these mathematical objects coverable by many open sets homeomorphic to Euclidean space; generalizations of surfaces.
Spivak's definition requires that they admit a metric, but this is not really necessary for most purposes.
manifolds