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I am reviewing the theory of manifolds, in particular vector fields, forms etc. Of course, in my opinion to understand the theory, I always try to find examples so as to illuminate it. For those things, when I consider sphere, it is more easier to understand by means of (computation, imagination).

But now, I would like to consider 1-dimensional real projective $M=RP^{1}$ space together with smooth structure that we always define.

Could you help me give a vector field on $M$ as a map: $M \longrightarrow TM$? I mean that I need exactly that map defined by that way, not in locally.

Or do you have another example vector fields on manifolds such that the ambient space is not Euclidean space and could you show it for me?.

The last thing I want to ask is that when we consider an n-dimensional manifold, can we have an algorithm to construct at least a non-trivial vector field on such a manifold?

Thanks.

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  • $\begingroup$ You can just always construct the 0-vector field given by $p \mapsto (p, 0_{T_pM})$ Or do you want something less trivial? $\endgroup$ – lush Feb 27 '18 at 20:06
  • $\begingroup$ Yep, I need an example that is not trivial. $\endgroup$ – PHU CUONG LE VAN Feb 27 '18 at 20:08
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    $\begingroup$ So what do you mean by "no in locally"? The thing is: you don't really understand $T_pM$ if you do not choose local coordinates. So I guess you'll want to choose those coordinates, otherwise it is really difficult to really explicitly describe elements in $T_pM$. Also: Do you want the vector field to be just merely any section of $TM \to M$, or do you want it to be continuous/smooth? $\endgroup$ – lush Feb 27 '18 at 20:39
  • $\begingroup$ Yep, I have constructed a vector field by using local charts. More precisely, we have only two local charts, then we can builts vector field in local in such a way that It can be glued together. $\endgroup$ – PHU CUONG LE VAN Feb 27 '18 at 21:01
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    $\begingroup$ They are always nonempty. it is not to hard to see that any constant map $M \to N$ is smooth. Furthermore $C^{\infty}(M)$ even is an $\mathbb{R}$-algebra, given by pointwise addition, multiplication etc. $\endgroup$ – lush Feb 27 '18 at 21:40
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Here is an example of an explicit family of vector fields on $\mathbb{RP}^1$. I will 1st give a coordinate version, then explain, without coordinates, where does this example come from.

Think of $\mathbb{RP}^1$ as th set of equivalence classes $[(x,y)]$ of non-zero vectors in $\mathbb{R}^2$, where two vectors are equivalent if one is a scalar multiple of the other. On the open subset of $\mathbb{RP}^1$ consisting of classes $[(x,y)]$ with $y\neq 0$, define the coordinate $t=x/y$. This is called an affine coordinate and is defined on the complement of the point $[(1,0)]\in\mathbb{RP}^1$. Another affine coordinate is $s=y/x$, defined on the complement of $[(0,1)]$. On the intersection of the two open subsets (coordinate charts) the two coordinates are related by $ts=1$.

Now define on the $t$ coordinate chart the vector field $q(t){d\over d t}$, where $q(t)$ is any polynomial function of degree not more than 2, i.e., $q(t)=at^2+bt+c$, where $a,b,c\in\mathbb{R}$. You can show that every such vector field, defined initially only on the 1st coordinate chart, extends uniquely and smoothly to all of $\mathbb{RP}^1$.

To do this, you can write this vector field in the 2nd coordinate chart, using the coordinate $s$, and find that the vector field is of the form $p(s){d\over d s}$, where $p(s)$ is also a polynomial of degree not more than 2, so extends smoothly to $s=0$ (the only point not covered by the 1st coordinate chart).

Where do these vector fields come from? They are the projectivization of linear vector fields on $\mathbb{R}^2$. These are vector fields of the form $(ax+by){\partial\over \partial x}+(cx+dy){\partial\over \partial y},$ where $a,b,c,d\in\mathbb{R}$. You can check that each such vector field defines a vector field on $\mathbb{RP}^1$, via the map $\mathbb{R}^2\setminus\{(0,0)\}\to\mathbb{RP}^1,$ $(x,y)\mapsto [(x,y)].$

You can generalize this easily to $\mathbb{RP}^n$, by writing explicit formulas as above in affine coordinate charts, for vector fields induced from linear vector fields on $\mathbb{R}^{n+1}.$

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