Coordinate Vectors Span Tangent Spaces
Given a point in an
-dimensional manifold
, we have the vector space
of tangent vectors at
. Given a coordinate patch
around
, we’ve constructed
coordinate vectors at
, and shown that they’re linearly independent in
. I say that they also span the space, and thus constitute a basis.
To see this, we’ll need a couple lemmas. First off, if is constant in a neighborhood of
, then
for any tangent vector
. Indeed, since all that matters is the germ of
, we may as well assume that
is the constant function with value
. By linearity we know that
. But now since
we use the derivation property to find
and so we conclude that .
In a slightly more technical vein, let be a “star-shaped” neighborhood of
. That is, not only does
contain
itself, but for every point
it contains the whole segment of points
for
. An open ball, for example, is star-shaped, so you can just think of that to be a little simpler.
Anyway, given such a and a differentiable function
on it we can find
functions
with
, and such that we can write
where is the
th component function.
If we pick a point we can parameterize the segment
, and set
to get a function on the unit interval
. This function is clearly differentiable, and we can calculate
using the multivariable chain rule. We find
We can thus find the desired functions by setting
Now if we have a differentiable function defined on a neighborhood
of a point
, we can find a coordinate patch
— possibly by shrinking
— with
and
star-shaped. Then we can apply the previous lemma to
to get
with . Moving the coordinate map to the other side we find
Now we can hit this with a tangent vector
where we have used linearity, the derivation property, and the first lemma above. Thus we can write
and the coordinate vectors span the space of tangent vectors at .
As a consequence, we conclude that always has dimension
— exactly the same dimension as the manifold itself. And this is exactly what we should expect; if
is
-dimensional, then in some sense there are
independent directions to move in near any point
, and these “directions to move” are the core of our geometric notion of a tangent vector. Ironically, if we start from a more geometric definition of tangent vectors, it’s actually somewhat harder to establish this fact, which is partly why we’re starting with the more algebraic definition.
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Just after “multivariable chain rule”, f(p) – f(0) should be f(x) – f(0).
p lives in the manifold rather than R^n.
Hi, I’m learning differential geometry and run into this blog, I must say you did a great job. But I have a question: “What does \circ means?”
It sounds like for some reason the TeX is not being rendered in your browser. \circ is the TeX code to generate the composition symbol
.