How do you prove immersion? (Basic Riemannian Geometry)

[b0it0i]

Homework Statement

the question i have is more of a conceptual question, i have no idea how to prove that a mapping would be an immersion. thus i have no clue how to start the assigned problem

here's the specific problem:
prove that (e~) is an immersion : note (e~ means phi tilda)

Let F: R^3--> R^4 be given by
F(x,y,z) = (x^2-y^2, xy, xz, yz), where (x,y,z) = p element of R^3

Let S^2 subset of R^3 be the unit sphere with the origin 0 element of R^3
Observe that the restriction (phi) e = F | S^2 is such that e(p) = e(-p),
and consider the mapping (phi tilda) e~: P^2(R) --> R^4 given by

e~( [p] ) = e(p)

where [p] is the equivalence class of p. [p] = {p, -p}

note that: P^2(R) means the real projective plane

Homework Equations

Definition of immersion: Let M^m and N^n be differentiable manifolds. A differentiable mapping
e: M --> N is said to be an immersion if
(dep = differential of e at p)
dep: TpM --> Te(p)N is injective for all p element of M

def of dep: Let M and N be differentiable manifolds. and let e: M --> N be a differentiable mapping. For every p element of M, and for each v element of TpM,
choose a differentiable curve a:(-E,E) --> M with a(0) = p, a'(0) = v.
Take B = e o a (phi compose a). The mapping
dep: TpM --> Te(p)N
given by dep(v) = B'(0)

The Attempt at a Solution

I thought you would need to use the definition of immersion, stated above, to show that a mapping is indeed an immersion. So i must show that

dep: TpM --> Te(p)N is injective for all p element of M.

HOWEVER: the examples in the book do not use this definition, or it skips over major steps, when they show that certain mappings are / aren't immersions.

for example:

1) The curve a: R--> R^2 given by
a(t) = (t, |t|) is not differentiable at t=0. a is not an immersion

2) The curve a(t) = (t^3 - 4t, t^2 - 4) is an immersion
a: R --> R^2
Do i need to use the definition of dep to find this curve "a".

def of dep: Let M and N be differentiable manifolds. and let e: M --> N be a differentiable mapping. For every p element of M, and for each v element of TpM,
choose a differentiable curve a:(-E,E) --> M with a(0) = p, a'(0) = v.
Take B = e o a (phi compose a). The mapping dep: TpM --> Te(p)N given by
dep(v) = B'(0)

can anyone clarify the concept / process to show how to prove a mapping is an immersion. I don't want the answer to this problem, just some guidance. By the way, this problem is not for any points, it's just a suggested problem.

 

[StatusX]

Well, it's pretty clear the function is differentiable (and so doesn't fail to be an immersion for the reason your first example does), so one way to proceed would be to compute its derivative and show its rank is 2.

 

[b0it0i]

I see, and you made that conclusion because of the Dimension theorem?

a linear function is 1-1 iff it's nullity is 0
iff rank = dim (P^2(R))