# Tagged Questions

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### Extending curves

I have the following situation, $N$ is a $k$-manifold, $X$ a compact $(k+1)$-manifold and $F:X \to N$ a smooth map. Let $y$ be a regular value of both $F$ and $F|_{\partial X}$, hence $F^{-1}(y)$ is a ...
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### Question about a specific case of the argument principle for maps of circles.

Problem Statement: Let $f:S^1\rightarrow S^1$ be a smooth map of manifolds where $S^1=\frac{[0,1]}{0~1}$, and let $f'(t)\in \mathbb{R}$ be given by the $df_t[1]_t=f'(t)[1]_{f(t)}$ at each $t\in S^1$. ...
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### Vector field on an odd sphere

Let $x^1,y^1,\ldots,x^n,y^n$ be the standard coordinates on $\mathbb{R}^{2n}$. The unit sphere $S^{2n-1}$ in $\mathbb{R}^{2n}$ is defined by the equation $\sum_{i=1}^n(x^i)^2+(y^i)^2=1$. Show that ...
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### An example of a differentiable manifold class $C^k$ but not class $C^{k +1}$

I'm looking for an example of a differentiable manifold of class $C^k$ but not class $C^{k +1}.$ I found an exercise in Hirsh's book, which suggests that the graph of $f (x) = |x|^{\lambda}$, where ...
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### Conditions for a projection of a Knot to be a Knot diagram.

friends. I'm working on a problem, the broad scope of which is to show that given a map $f:S^1\rightarrow \mathbb{R}^3$ be a smooth embedding, and a projection map $\pi_v:S^2\rightarrow P_v$, where ...
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### Tangent space to a product

Can you explain this question explicitly. This is a little bit difficult for me, but I want to learn how to solve. Thank you for help. If $M$ and $N$ are manifolds, let $\pi_1:M\times N\to M$ and ...
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### Non-degenerate smooth functions on a manifold

I am trying to prepare a presentation on "the use of differential geometry in the theory of critical points", but only the case where there is a single variable. (only in dimension 1), and i ask ...
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### smoothness of hopf fibration projection with respect to standard differentiable structure on unit sphere

We know that steographic projection defines a differentiable structure on $S^n$ by sending points on $S^n$ to hyperplane $\{x^{n+1}\}=0$. In fact, stereographic projection $\sigma_P: S^n- P\to R^n$ ...
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### Level Sets are Regular Submanifolds

In Section $9$ of Tu's Introduction to Manifolds, we're asked to find all values $c\in\Bbb R$ for which the level set $f^{-1}(c)$ is a regular submanifold when $$f(x,y)=x^3-6xy+y^2.$$By taking each of ...
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### Write down the equation of the tangent plane and compute the Taylor series of the function

Set $f(x,y,z) = x + y + z + x^2 + y^2 + z^2$. Consider the surface $$S = \{f(x,y,z) = 0\} \subset \mathbb{R}^3$$ near the origin $o = (0,0,0) \in S$. Write down the equation of the ...
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### A question about orthogonal projections of Smooth Embeddings of the circle.

Question: Let $f$ be a smooth embedding of $S^1\rightarrow \mathbb{R}^3$. Given an element $v\in S^2$ we have the orthogonal projection $\pi_v:\mathbb{R}^3\rightarrow P_v$ to the plane $P_v$ = the ...
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### Let $G$ be a Lie group. Show that there is a diffeomorphism $TG \cong G \times T_e G$.

Since $T_p G$ is isomorphic to $T_e G$ for all $p\in G$, it makes sense that each vector in $T_p G$ can be identified with a vector in $T_e G$. Hence, to make the map from $TG$ one to one, we must ...
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### Tangent Bundle of a Riemann Surface

Let $g$ be the genus of a closed Riemann surface, what can be said about $g$ if the tangent bundle $T$ of that surface is trivial? From the formula for the degree of a tangent bundle, $\deg(T)=2-2g$, ...
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### Questions (doubts) on: Group Action on Manifolds

There are 2 questions that are bugging me in differential topology and I'd be glad if the same could be cleared up: Let $X = x\frac{\partial}{\partial y}$ be a vector field on $M = R^2$, where $R$ ...
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### Image of smooth manifold is a submanifold

It's know that if $M$ is a compact, smooth manifold of dimension $n$ and the map $f: M \to \mathbb{R^m}$ is injective, smooth, $n \le m$ and $Jf(a)$, the Jacobian, has rank $n$ for every $a \in M$, ...
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### $f$ a differentiable map between manifolds of same dimension; $df(p)$ is nonsingular - show $f$ is an open map

Let $f: X \to Y$ be a differentiable map of manifolds where $dim \;X = dim\;Y = n$. If $df(p)$ is nonsingular for all $p \in X$, show $f$ is an open map. So here is what I was thinking: As $df(p)$ ...
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### Compute $\chi(\mathbb{C}\mathrm{P}^2)$.

I need to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using techniques from differential topology. I cannot think of any theorems that are particularly useful for this computation, so I think that I will ...
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### Commuting smooth maps

Suppose $f:A\to B$ and $g:C\to D$ are smooth embeddings, $h:B\to D$ is a smooth map, and $i:A\to C$ is a continuous map such that $g(i(x))=h(f(x))$. Then, how to show that $i$ is smooth? An ...
This is a homework assignment. I'm not sure I even understand the question fully, as the parametrization seems slightly wrong. Over an $n$-ball, let $r \in [0,1)$ denote a radial coordinate, and let ...
### If $S^{2n+1}$ is covering space of $X$, then $X$ is orientable.
Is there any direct way to prove that $n$-manifold is orientable? In AT we can just calculate $n$'th homology group and check whether it's $\mathbb Z$ or $0$. But I want a geometric method, using ...