Topology _the nurbs surface_Manifolds_Riemannian ManifoldsTo measure distances and angles on manifolds, the manifold must beRiemannian. A Riemannian manifold is a differentiable manifold inwhich each tangent space is equipped with an inner product 〈⋅,⋅〉in a manner which varies smoothly from point to point. Given twotangent vectors u and v, the inner product 〈u,v〉 gives a realnumber. The dot (or scalar) product is a typical example of an innerproduct. This allows one to define various notions such as length,angles, areas (or volumes), curvature, gradients of functions anddivergence of vector fields.
Topology _the Klein Bottle_ A Klein Bottle is a 4-Dimensional topography that cannot beembedded within 3-Dimensional space. The surface has somevery interesting properties, such as being one-sided, like theMoebius strip; being closed, yet having no "inside" like a torusor a sphere; and resulting in two Moebius strips if properly cutin two.