Future Technology Circuits International

Manifold

Manifolds (or differential manifolds) are one of the most fundamental concepts in mathematics and physics. A primitive definition of a manifold corresponds to a space that maybe curved and have a complicated topology, but in local regions looks just like $$\textbf{R}^n$$. In other word, a manifold is a space consisting of patches that look locally like $$\textbf{R}^n$$, and are smoothly sewn together. A manifold can also be defined as a set that has the local differential structure of $$\textbf{R}^n$$ but not necessarily its global properties. However, a precise formulation of the concept of a manifold requires some preliminary definitions.

• Open ball

An open ball of radius $$r$$ is the set of all points $$x$$ in $$\textbf{R}^n$$ such that $$|x-y| < r$$ for some fixed $$y\in \textbf{R}^n$$ and $$r\in \textbf{R}$$, where $|x-y|=\left [\sum^n_{\mu=1} (x^\mu-y^\mu)^2\right]^{1/2}.$

• Open set

An open set in $$\textbf{R}^n$$ is any set which can be expressed as a union of open ball. This notion of open set makes $$\textbf{R}^n$$ a topological space.

• Chart

A chart or coordinate system consists of a subset $$U$$ of a set $$M$$ along with a bijective map $$\phi:U\longrightarrow \textbf{R}^n$$, such that the image $$\phi(U)$$ is open in $$\textbf{R}^n$$.

• Atlas

A $$C^\infty$$ atlas is a family of charts $$\{(U_\alpha,\phi_\alpha)\}$$ which satisfies two following conditions:

1- The $$U_\alpha$$ cover $$M$$, $$i.e.$$ $$\bigcup_\alpha U_\alpha =M.$$

2- If $$U_\alpha \bigcap U_\beta$$ is non-empty, then the map $$\phi_\alpha \circ \phi_\beta^{-1}:\phi_\beta(U_\alpha \bigcap U_\beta)\longrightarrow \phi_\alpha(U_\alpha \bigcap U_\beta)$$ is a $$C^\infty$$ map of an open subset of $$\textbf{R}^n$$ to an open subset of $$\textbf{R}^n$$ (see figure 1).

• Manifold

A $$C^\infty$$ n-dimensional manifold is a set $$M$$ together with a $$C^\infty$$ atlas $$\{(U_\alpha,\phi_\alpha)\}$$ so that the above conditions are satisfied.