Proof that a specific map is an injective immersion

[jacobrhcp]

Homework Statement

Consider f: R^{m+1} - {0} -&gt; R^{(m+1)(m+2)/2}, (x^{0},...,x^{m}) -&gt; (x^{i} x^{j}) i<j in lexicografical order

a) prove that f is an immersion
b) prove that f(a) = f(b) if and only if b=±a, so that f restricted to Sm factors through an injective map g from Pm.
c) show g is an embedding

The Attempt at a Solution

for a), (x^{i} x^{j}) is just a one-dimensional image? I think this because it is just the product of two specific components of (x^{1},...,x^{m}), and hence just one-dimensional. But then why does the question state f is sent to some high power of R?
Continuing, under the assumption I was right, something is an immersion if its derivative matrix is injective and it's a map between manifolds. In this case the derivative matrix is just a (m+1)x1 vector, because the image is one-dimensional. Most components are zero, except the derivatives of x^i x^j with respect to x^i and x^j. Because Df(a)=Df(b) if and only if a=b, this would be an injective map. Can anyone check this and tell me if it's half right, I don't feel very confident about it?

for b), I'd say this map is not injective at all since we just kill all components except x^i and x^j, so we could change any component except the ith or jth and still be the same f(a). I must have understood something wrong.

c) was easy. A theorem states that a closed injective immersion is an embedding. because any injective immersion from a compact domain is closed (as in the case with g and Pm), g is an embedding. I also proved Pm was compact, because it is the continuous image of Sm, which is very compact in Rn.

 

[Mark44]

Homework Statement

Consider f: R^{m+1} - {0} -&gt; R^{(m+1)(m+2)/2}, (x^{0},...,x^{m}) -&gt; (x^{i} x^{j}) i<j in lexicografical order

a) prove that f is an immersion
b) prove that f(a) = f(b) if and only if b=±a, so that f restricted to Sm factors through an injective map g from Pm.
c) show g is an embedding

The Attempt at a Solution

for a), (x^{i} x^{j}) is just a one-dimensional image? I think this because it is just the product of two specific components of (x^{1},...,x^{m}), and hence just one-dimensional. But then why does the question state f is sent to some high power of R?
Continuing, under the assumption I was right, something is an immersion if its derivative matrix is injective and it's a map between manifolds. In this case the derivative matrix is just a (m+1)x1 vector, because the image is one-dimensional. Most components are zero, except the derivatives of x^i x^j with respect to x^i and x^j. Because Df(a)=Df(b) if and only if a=b, this would be an injective map. Can anyone check this and tell me if it's half right, I don't feel very confident about it?

for b), I'd say this map is not injective at all since we just kill all components except x^i and x^j, so we could change any component except the ith or jth and still be the same f(a). I must have understood something wrong.

c) was easy. A theorem states that a closed injective immersion is an embedding. because any injective immersion from a compact domain is closed (as in the case with g and Pm), g is an embedding. I also proved Pm was compact, because it is the continuous image of Sm, which is very compact in Rn.

Can you clarify for me the definition of this function? The first part of the definition indicates that it's a map from an (m + 1)-dimensional space to a space of dimension (m + 1)(m+2)/2. The notation used in the definition of this function doesn't convey the idea that the image of this map has dimension (m + 1)(m+2)/2.

It's also not clear to me what the image of a specific input looks like. For starters, let's assume that m = 2, in which case the map is from R^3 to R^6. I think the part of the definition that shows what results from a given input means that f carries a triple of numbers, ( x^0, x^1, x^2) to a 6-tuple (I'm still assuming m = 2) of numbers: (x^0 * x^1, x^0 * x^2, x^1 * x^2, x^2 * x^0 , x^2 * x^1, x^1 * x^0).

The first three products of the six I've listed satisfy the i < j condition in the definition, but the last three don't, so maybe their factors could be reversed to give this 6-tuple:
(x^0 * x^1, x^0 * x^2, x^1 * x^2, x^0 * x^2 , x^1 * x^2, x^0 * x^1).

Can you help me understand the given information in this problem?

 

[jacobrhcp]

I wrote down the definitions exactly as in the question, except that my 'R' is supposed represent the real R.

So unfortunately, I cannot. I am as confused as you are, and frankly I was hoping you could understand the definition better than I could.

the text of the question litterally is:

Consider a map f:R^(m+1) - {0} -> R^((m+1)(m+2)/2) which assigns to (x^0,...,x^m) in R^(m+1) the vector (x^i x^j), i smaller or equal to j (in lexicographical order, say)

a) Prove f is an immersion
b) prove f(a)=f(b) if and only if b=±a, so that f, restricted to Sm factors through an injective map g:Pm -> R^((m+1)(m+2)/2)
c) prove that g is an embeddingexcept in the question they had an actual smallerequal-sign, and the R was in handwritten bold, the real space.

if you don't trust my copying, you can find the question yourselves on the bottom of page 16 of this set of lecture notes:

https://www.math.uu.nl/people/looijeng/difftop06eng.pdf
*the question was modified in c slightly because the teacher admittedly made a typo there.Also, reading your suggestion... I think you are right probably. It is seems the most plausible of things so far, and things with i's or j's are oftenly summed over or repeated for no apparent reason without notice... even when einsteinconvention is in the back of my mind.

I'm going to work with that now.

 

[jacobrhcp]

well... I think it's as good as it's going to get. So for your inspiration and attempted help, I thank you.

If I use you definition I showed everything. If you want I can show how, but I don't really feel like it, so you'll have to specifically ask for it. Solved as far as I'm concerned! (how do you mark your thread 'solved' again?)

 

[Mark44]

well... I think it's as good as it's going to get. So for your inspiration and attempted help, I thank you.

If I use you definition I showed everything. If you want I can show how, but I don't really feel like it, so you'll have to specifically ask for it. Solved as far as I'm concerned! (how do you mark your thread 'solved' again?)

I took a look at the document you provided, and I understand things a little bit more. Not a whole lot, but a little.
First, the map is from R^(m+1) - { 0 }. That wasn't clear to me in the original post.
Second, I think I understand what the image vectors look like. On p. 17 of the document, in Remark 4.9, he refers to the problem you're working on, using m = 2, as I did.

In that example, the map carries a triple (x^0, x^1, x^2) in R3 - {(0, 0, 0)} to a 6-tuple (1*1, 1*x, 1*x^2, x*x, x*x^2, x^2*x^2). This simplifies to (1, x, x^2, x^2, x^3, x^4). Looking at the example he gives, I think he meant i <= j, not i < j, in the ordering of the coordinates of the image vector.

That's about all I can do, since I don't know much about manifolds and differential geometry. I hope I've been able to tell you something you didn't already know.