Tangent Spaces and Regular Values
If we have a smooth map and a regular value
of
, we know that the preimage
is a smooth
-dimensional submanifold. It turns out that we also have a nice decomposition of the tangent space
for every point
.
The key observation is that the inclusion induces an inclusion of each tangent space by using the derivative
. The directions in this subspace are those “tangent to” the submanifold
, and so these are the directions in which
doesn’t change, “to first order”. Heuristically, in any direction
tangent to
we can set up a curve
with that tangent vector which lies entirely within
. Along this curve, the value of
is constantly
, and so the derivative of
is zero. Since the derivative of
in the direction
only depends on
and not the specific choice of curve
, we conclude that
should be zero.
This still feels a little handwavy. To be more precise, if and
is a smooth function on a neighborhood of
, then we calculate
since any tangent vector applied to a constant function is automatically zero. Thus we conclude that . In fact, we can say more. The rank-nullity theorem tells us that the dimension of
and the dimension of
add up to the dimension of
, which of course is
. But the assumption that
is a regular point means that the rank of
is
, so the dimension of the kernel is
. And this is exactly the dimension of
, and thus of its tangent space
! Since the subspace
has the same dimesion as
, we conclude that they are in fact equal.
What does this mean? It tells us that not only are the tangent directions to contained in the kernel of the derivative
, every vector in the kernel is tangent to
. Thus we can break down any tangent vector in
into a part that goes “along”
and a part that goes across it. Unfortunately, this isn’t really canonical, since we don’t have a specific complementary subspace to
in mind. Still, it’s a useful framework to keep in mind, reinforcing the idea that near the subspace
the manifold
“looks like” the product of
(from
) and
, and we can even pick coordinates that reflect this “decomposition”.