A projective plane diffeomorphism is a smooth bijective mapping between two projective planes that preserves the smooth structure of the surfaces. This means that the mapping is differentiable and its inverse is also differentiable.
SO(3) is a special group of 3-dimensional rotations in Euclidean space. It stands for Special Orthogonal group in 3 dimensions and is also known as the rotation group. It is a continuous group, meaning it contains infinitely many elements.
A subset of SO(3) is a collection of elements from the SO(3) group that satisfies certain properties. In this case, the subset is a collection of rotations that form a projective plane.
To show that a subset of SO(3) is diffeomorphic to a projective plane, we need to find a smooth bijective mapping between the two surfaces. This can be done by identifying the elements of the subset with the points on the projective plane and showing that the mapping preserves the smooth structure of both surfaces.
Showing that SO(3) subset is projective plane diffeomorphic is significant because it helps us better understand the relationship between these two mathematical concepts. It also allows us to apply concepts and techniques from one domain to the other, leading to new insights and developments in both fields.