— 1. Some objects on the manifolds —
— 1.1. Curves and functions —
When I was in the beginner physics class, I often analyzed the track of a particle on a plane that moves under some potential. Ordinarily, the curve on the plane can be parameterized using one parameter, say
, such that every points on the curve can be marked with some definite value of this parameter
. We also can make an arbitrary choice to place the zero of
; nature will not know which point you identify with the
. If I make a coordinate system on this plane (just imagine for simplicity that this is a two-dimensional plane and we use Cartesian system for the coordinate), then a point on the curve has the coordinate

such that if you have a single number
, then you will have two numbers, one for
and one for
, and it gives you the precise position of the point on the curve (this is why we often say that curve is one-dimensional). You can make an arbitrary curve on this plane which touches every parts of this plane, and by principal, you always can identify its points using this single parameter
.
However, this advantage to use the powerful parameter
to mark points on the curve cannot hold peacefully if we make a generalization from the plane to a manifold. The problem is, we cannot make a global coordinate system which covers all parts of manifold. Instead, we need to define some local coordinate systems, provided that we still can transform each other using continuous mapping. Therefore, the principal that we can identify the points on the curve by using coordinate
doesn’t make sense anymore, since two neighborhoods on the manifold have different convention about the values of
and
.
To solve this problem, we need to make two mappings. First, consider the mapping
such that
whenever
. We can imagine that this is the curve which connects two points on the manifold, namely
and
, and it is a simple one; there is no intersection of this curve with itself (this assumption is just for the sake of simplicity). Of course you don’t need to have finite numbers for
and
, the two queens of pain (
and
) will also work. While the first mapping is the one which maps
to
, the second mapping is
, i.e. the mapping that serves as the coordinates marker for each points on the manifold. Therefore, we can construct the composition mapping
, such that from a single parameter
you will get
values of coordinates.
If the part of
is mapped to
and produces what we intuitively call as curve, then the mapping from
to
makes a very different object: this is simply the ordinary function defined on a manifold. Concretely, if we have a function
, and take a point
with the coordinate
, then we will have the composition mapping
. In physics, we often encounter this composition mapping
, which relates the coordinates of points on the space with a value in each points. Actually, this physicist’s view will not make any trouble if we can define a global coordinate system on the manifold, but if we just could make a system of local coordinates, then we need to distinguish the role of
, the function from points of the manifold to some real numbers, and the role of
, the attachment of points to a local coordinate system.
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