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b0it0i

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## 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.

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