Coordinates - Map Projections: Cartographic Information Systems - page 129

3-1 Coordinates Relating to Manifolds

Coordinates relating to manifolds, in particular, differential manifolds, elements of topology
(Hausdorff topological space, open and closed domains).

Let us here first consider manifolds and their charts, the complete atlas, and the minimal atlas.
Let us here first define the term chart according to Definition 3.1 .

Definition 3.1 (Manifold, chart).

n is locally an n -dimensional Hausdorff topological space .Thatisto

An n -dimensional manifold M

n , called “point”, there is a connected open neighborhood

any element of M U and a homeomorphism

n n is locally homeomorph to n . Any homeomorphism

Φ : U → φ ( U )toanopenset φ ( U )of R · M R

n . A one-dimensional manifold is called curve , a two-dimensional manifold
with Riemann metric and without Cartan torsion is called surface , and a compact, connected
manifold without boundary is called closed manifold .

φ is called a chart of M

End of Definition.

Indeed, we implemented many unknown notions from the theory of morphism , in particular, from
topology, but we do not hope to lose you, the map maker. Therefore, just follow us to stroll along
Hausdorff Street, Bonn (Germany) to meditate over Haussdorff's axiom of separation (T 2 ) within
Listing's topoploy , shortly reviewed in the Appendix. What is more important here is the question
that follows.

Question: “Why has the notion of the manifold and
the chart, ), been axiomatically introduced?”

Answer: “The answer to our fundamental question is based
on the “mathematical observation” that not all higher-
dimensional curved surfaces can be embedded or immersed
in a higher-dimensional Euclidean space. Or we do not
know whether such an embedding or immersion exists.”

U → φ ( U

For instance, since the twenties of the 20th century, we know that spacetime is a four-dimensional
pseudo-Riemann manifold of signature “+ + + ” equipped with a pseudo-Riemann metric, Rie-

mann curvature, zero Cartan torsion. But how to embed or immerse such a four-dimensional
spacetime manifold in a pseudo-Euclidean space? In case we know nothing, we better work
“intrinsically” with the manifold, neglecting the problem of embedding or immersion. Indeed,
this is the majority vote procedure when dealing with spacetime. And, in addition, it would
not be too helpful to think in terms of the following theorem: any analytical four-dimensional
pseudo-Riemann manifold (analytical spacetime) can be immersed in a ten-dimensional pseudo-
Euclidean space. Another example is the projective space

−

n , and this projective space cannot be
embedded or immersed in an Euclidean space as the ambient space. Anyway, let us continue to
explain continuity and homeomorphism , a situation similar to art, where many “. . . isms” exist.
At least, the mathematical builders of the world are more careful in defining their “. . . isms”,
like conformeomorphism or areomorphism .

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