Calabi Yau Manifold
A Calabi Yau Manifold is a Complex surface on which we can assign a Ricci-flat (Kahler) metric. This has been shown to be equivalent to it having \(SU(n)\) holonomy (this describes the way arrows are transformed when moved in a loop). All smooth algebraic varieties in \( \mathbb{CP}^n\) are Kahler manifolds.Compact dimensions
We shall give some examples of Calabi Yau manifolds. One of the the most symmetrical is: $$ \{(v,w,x,y,z)\in \mathbb{CP}^4 | v^5+w^5+w^5+x^5+y^5+z^5 = 0 \}$$ (This is equivlanet to the surface )Mistake here as indices for CP4 are 0..4 A member of this type therefore includes a member of \( \mathbb{CP}^4 \) together with a proof that it satisfies the equation.
Projective Space
A projective space over field \(F\) is defined by: $$ F\mathbb{P}^{n} \cong F^{n+1}\backslash \{0\} / \{ \forall_{\alpha\in F}(x_1,..x_{n+1}) \sim (\alpha y_1,..,\alpha y_{n+1}) \} $$(Not sure how to do equivalence relations yet)
Isomorphisms
Some isomorphisms between different types (topological spaces) are:Also: $$\mathbb{ZP}^1\backslash\{(1,0)\} \cong \mathbb{QP}^1\backslash\{(1,0)\} \cong \mathbb{Q}$$
Homeomorphism
The definition of homeomorphism could be something like: $$A\cong B \rightarrow \exists_{\phi:A\rightarrow B, \psi:B\rightarrow A} (\forall_{y:B}\exists_{x:A}\phi(x)=y \wedge \psi(\phi(x))=x ) \wedge (\forall_{y:A}\exists_{x:B}\psi(x)=y \wedge \phi(\psi(x))=x ) $$ With additional conditions that \(\psi\) and \(\phi\) be continuous.For example for the circle we might have \( \phi(x,y) = atan2(y,x) , \psi(\theta)= (\cos(\theta),\sin(\theta)) \)