# Tangent bundles and $\mathbb{C}P^n$ and $\mathbb{C}^n$

As discussed here the complex projective space $\mathbb{C}P^n$ is the set of all lines on $\mathbb{C}^n$ passing through the origin. It would seem natural to assume that any $\mathbb{C}P^n$ can be viewed as a tangent space to a point in an equivalently parametrized space. So for a point $p$ on some manifold $M$ the set of all tangent vectors at $p$ is:

$$T_pM$$

As discussed here, the union of all points on a manifold is written as:

$$TM = \bigcup_{p\in M} T_pM$$ and is known as the tangent bundle.

Also, $\pi$ is used to represent the map of from the tangent bundle to the manifold such that $\pi: TM \rightarrow M$ and for each $p \in M$ :

$$\pi^{-1}(p)=T_pM$$

Where $\pi^{-1}(p)$ is a vector space of dimension $n = \dim M$ (confirming as stated above that the tangent space will have same dimension, or equivalent number of parameters as the underlying manifold).

Ordinary spacetime is often described as $\mathbb{R}^{3,1}$ and it is known that euclidean plane $\mathbb{R}^{2}$ can be described by a complex number in $\mathbb{C}^{1}$, and $\mathbb{R}^{1,1}$ can be described with split-complex numbers in $\mathbb{R}^1\oplus \mathbb{R}^1$. When one considers the Lorentz boost, it is tempting to think of ordinary space as being $\mathbb{C}^{1}\oplus \mathbb{R}^1\oplus \mathbb{R}^1$ and described by two variables $\left( z , \mathring{z} \right)$ where $z$ is a complex number and $\mathring{z}$ is a split complex number.

Since there are exactly three $2$-dimensional unital algebras, complex numbers, split-complex numbers and dual numbers, and the algebra of dual numbers is isomorphic to the exterior algebra of $\mathbb{R}^1$, and there are two separate $\mathbb{R}^1$ in $\mathbb{C}^{1}\oplus \mathbb{R}^1\oplus \mathbb{R}^1$, it is tempting to add an additional set of dual numbers so that we can parameterize space as $$\left( z , \mathring{z} \right)\big| \left( \mathring{v} , \mathring{w} \right)$$ where $\mathring{v}$ is the dual number for the first $\mathbb{R}^1$ and $\mathring{w}$ is the dual number for the second $\mathbb{R}^1$. Dual numbers are noteworthy as having "fermionic" directions of "bosonic" directions.

If I wanted to keep my new set of coordinates, $$\left( z , \mathring{z} \right)\big| \left( \mathring{v} , \mathring{w} \right)$$ how best would I describe the tangent space of this "ad hoc" manifold? Would I be able to describe this in $\mathbb{C}P^n$, if not, why not? If only because of compactness, why can't tangent spaces be compact? Could we describe it as pseudo-tangent?

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$\mathbb{C}P^n$ will not be a tangent bundle... it is compact whereas tangent bundles are not. And your description of the tangent bundle is missing crucial features, the way the $T_pM$ "connect" to each other. That all being said, I have no idea what the latter half of your post has to do with the beginning. Please elaborate and just delete the first part about tangent bundles (providing instead a link to Wikipedia that defines it) –  Chris Gerig Mar 3 at 1:50
@ChrisGerig thanks for the input, I added a reference, and some additional questions, I will revisit this later for further modifications. –  Hal Swyers Mar 3 at 2:00
Comment to the question formulation (v5): What does $\big|$ mean? –  Qmechanic Mar 4 at 1:09
@Qmechanic Nothing in particular, however, I wanted to group the coordinates appropriately and so I borrowed $\big|$ from the idea of superspace notation. –  Hal Swyers Mar 4 at 1:31
Comments on the question (v5), which has been flagged for being math only. OP writes: Why can't tangent spaces be compact? The answer is: Because tangent spaces are by def. vector spaces. Perhaps OP wants a fiber bundle construction rather than a vector bundle? Is the word pseudo-tangent supposed to be a standard mathematical notion, or is it just supposed to be a yet-to-be-seen-unknown-notion that replaces the word tangent? Why is OP talking about the tangent bundle of Minkowski space, when he really wants to ask about $\mathbb{C}P^n$? Is it possible to make the question more clear? –  Qmechanic Mar 5 at 15:27