For ten points, identify this non-orientable manifold, the "neck" of which self-intersects and attaches to a hole in the "base."
This is a two-dimensional differentiable manifold without an unambiguous normal. It can be constructed mathematically by taking the connected sum of two real projective planes, and it cannot be embedded in Euclidean 3-space, though it can be immersed there. Like a torus, it is bounded by a compact closed surface that has an Euler characteristic of zero. It is equivalent to two Möbius bands joined along their boundaries. For ten points, identify this non-orientable manifold, the "neck" of which self-intersects and attaches to a hole in the "base."
This is a two-dimensional differentiable manifold without an unambiguous normal.
Klein bottle