文章与日志

Basic Notions of Manifold

When we mention Euclid space, maybe it induct excess meanings. So,
we can try to construct an object that its meaning is less than
Euclid space but however, is meaningful and analyzable.

At least, we need some points, then, if we want a point is
analyzable, it must has its neighborhood, moreover, to make it
computable or realizable, let its neighborhood homeomorphism with
an open set of Euclid space.

Such points constitutes a set, then make this set is a separated
set, a Hausdorff set, we definite such a set a manifold.

The point `x\in M`, M is a manifold. U is a neighborhood of x, and
there is a homeomorphism `\phi:U\mapsto \phi(U)`. then, Any such a
$(U,\phi(U))$ means a realization of any point of U: $\phi(y)$ is
the coordination of $y\in U$.

So, we definite $(U, \phi(U))$ a map chart of M.

Because any point of a manifold has its neighborhood as well as an
homeomorphism, then for any pair of points (x, y), their
neighborhood (U, V) have intersection or have not:

i) if $U\cap V\neq\theta$, then $\phi(U)\circ\varphi(V)^{-1}$ and
$\varphi(V)\circ\phi(U)^{-1}$ are two real functions from an open
set to another of Euclid space. we can treat them as coordinations
transformation, we can let them be $C^{r}$.

ii) if $U\cap V=\theta$, then nothing.

When i) and ii) are satisfied, we name U and V are $C^{r}$
consistent.

If we want to concern the movement of point on a manifold, and to
analyze such movement, we'd better let the functions
$\phi(U)\circ\varphi(V)^{-1}$ and $\varphi(V)\circ\phi(U)^{-1}$
have good enough analytical property, because their analytical
property restrict the analyzable of the manifold.

Then, as soon as we get enough map charts $(U, \phi(U))$ of
manifold M, we can analyze the point movement on it. the "enough"
means:

i) $\mathbbm{A}=\{U, V, W,...\}$ is a open cover of M;

ii) Any two of $\mathbbm{A}$ are $C^{r}$ consistent;

iii) $\mathbbm{A}$ is the maximum.

then, we name $\mathbbm{A}$ a $C^{r}$ differential structure of M,
correspondingly, M is a $C^{r}$ differential manifold.

Let's treat some familiar set as manifold, and to see how does
adding various differential structures on them means.
Euclid space $R^{n}$
Sphere $S^{n}$
Projective space $P^{n}$
Milnor monstrous ball.