The Projective Group as a Topological Manifold

In this article, we start by a review of the circle group  [1] and its topology induced [1] by the quotient metric, which we later use to define a topological structure on the unit circle ( ) 1 1,τ  . Using points on 1  under the complex exponential map, we can construct orthogonal projection operators. We will show that under this construction, we arrive at a topological group, denoted [ ] ( ) P G θ of projection matrices. Together with the induced topology, it will be demonstrated that [ ] ( ) P G θ is Hausdorff and Second Countable forming a topological manifold. Moreover, I will use an example of a group action on [ ] ( ) P G θ to generate subgroups of [ ] ( ) p G θ .


Introduction
Orthogonal Projection is a very familiar topic in Linear Algebra [2].With reference to [2], it is already known that if V is a finite-dimensional vector space and P is a projection on W V ⊂ , where W is a subspace of V. Then P is idempotent, that is 2 P P = .P is the identity operator on W, that is : x W Px x ∀ ∈ = .We al- so know that W is the range of P and if U is the kernel of P then , , V W U w W u U = ⊕ ∈ ∈ .It is easy to show that ( ) . It is also a known fact that these operators are bounded i.e.Px x ≤ . In this paper we will focus on projections in 2

Notation Used in This Article
1) ( ) The unit circle as a Topological Group.2)  The circle group defined as : / 2π =    .
3) [ ] θ is an element in the topology of  . 4)τ  the usual topology on  .5) 1 τ  the topology on the unit circle 1 where [ ] { } It is clear that this is associative and hence  is a group.

The Topological Structure on 𝕋𝕋 [1] [3]
First, we see that we have the projection mapping such that For the intersections we get ( )

The Quotient Metric on the Circle Group
The quotient topology τ  is induced by the quotient metric defined as We can define an open ball from this metric in the following way θ ′ be some other point in  .Then we think of . Hence, the open sets in  can be defined by using the definition of the open balls (above) and the canonical projection mapping π .
In order for the canonical map π to make sense and in order to satisfy This defines a topology on  .Furthermore, even though Also we note that this metric is a pseudo-metric.A pseudo-metric is metric is similar to usual metric spaces with the exception that it possible to have the fol- Hence,  is a topological group, denoted ( )

:   →
We now consider the unit circle as the set Clearly, 1  can e endowed with the subspace topology 1 τ  generated by the metric [ ) Clearly, this defines a topology on 1  .Equipped with this topology we can say that ( ) This, clearly defines an isomorphism.The group operation on the circle is given by multiplication as follows Given that each open set in the topology satisfies the metric ( ) implies that Φ is bijective and hence has an inverse.

Moreover, we have
This is just the familiar projection formula.Hence, the image is just the The Kernel, substitution of u θ by gives us the following result ( ) Hence, the kernel is the orthogonal complement of the subspace spanned by

Projections as a Topological Manifold
Clearly, the topology ( ) . Hence, by the group homomorphism we ( ) ( ) ( ) Therefore, it is Hausdorff.
For the second countability property, we proceed as follows.
Starting by using a countable basis in  of the form ( ) , θ θ′ where , θ θ′ ∈  .Since  is then the set open and bounded in  , any open set in  can be written as the countable union of open balls ( ) , θ   .
such that we define metric to be ( )e ,e i i d θ θ′ i.e.the shortest arc length between the points ( )


It is clear that open balls on  are mapped into open arcs on 1 .We now focus on the topology generated by open arcs.We can write ( ) Is a countable basis on  .This implies that the mapping { } exp i induces a second countable basis on ( )