How do you prove immersion? (Basic Riemannian Geometry)

In summary: 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 bydep(v) = B'(0)has rank 2 because it takes two differentiable curves to get from (-E,E) to M.
  • #1
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.
 
Last edited:
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  • #2
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.
 
  • #3
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))
 

1. What is immersion in Riemannian Geometry?

Immersion in Riemannian Geometry refers to a differentiable mapping from a manifold to a higher dimensional space, where the first fundamental form is positive definite. It is a way to embed a lower dimensional space into a higher dimensional one.

2. How do you prove that a mapping is an immersion?

To prove that a mapping is an immersion, you need to show that the derivative of the mapping is injective at each point on the manifold. This means that the tangent vectors at each point are linearly independent, which ensures that the mapping preserves the local geometry of the manifold.

3. What is the importance of proving immersion?

Proving immersion is important in Riemannian Geometry because it allows us to study a lower dimensional manifold by embedding it into a higher dimensional space. This can help us understand the properties and structure of the manifold in a more accessible way.

4. Can a manifold have multiple immersions?

Yes, a manifold can have multiple immersions. This is because there can be different ways to embed a lower dimensional manifold into a higher dimensional space while preserving its local geometry. However, there is always a unique immersion called the canonical immersion, which is the one that minimizes the distortion of the manifold.

5. Are there any applications of immersion in real-world problems?

Yes, immersion has many applications in various fields such as computer graphics, robotics, and physics. In computer graphics, immersion is used to create realistic 3D models of objects by embedding their 2D images. In robotics, immersion is used to map the environment and plan robot movements. In physics, immersion is used to study the behavior of particles in a curved space-time.

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