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Questions tagged [differential-topology]

Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

7287 questions
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Interior of image of regular points is dense?

I'm studying some problems related to differential topology and I came across the following exercise: if $f:M\rightarrow N$ is a surjective smooth (i.e., $C^\infty)$ function, $\dim(M)>\dim(N)$ and $R_f$ is the set of regular points of $f$, then the…
Renan Mezabarba
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continuity of exterior derivative

Stimulated by an exercise in da Silva: For a time-dependent vector field $v:M\times \mathbb{R}\to TM$, $k$-form $w:M\to T^{k,0}{M}$ and the isotopy $\rho:M\times \mathbb{R} \to M$ generated by $v$, show that $\dfrac{d}{dt} \rho^*_t \omega=\rho^*_t…
cjackal
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Differential topology, fundamental theorem of algebra

I am reading Milnor's Topology from the Differentiable Viewpoint, in particular page 8-9 about applying regular values to prove the fundamental theorem of algebra. So he defines stereographic projection as $h_{+}: S^{2}-\{(0,0,1)\}\to…
Rene Cabrera
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Question about definition of vector field

The following definition of vector fields were given in class, A $\textbf{Smooth Vector Field on a manifold $M$}$ is a smooth function, $$s: M \to TM$$ such that $$\pi \circ s=id_M$$ We then went on to say that there are two alternative…
Yuugi
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Interpretation of local submersion theorem

Wikipedia gives the following formulation of the $\textbf{Local Submersion thoerem}$, If $f: M \to N$ is a submersion at $p$ and $f(p)=q \in N$, then there exists an open neighborhood $U$ of $p$ in $M$, an open neighborhood $V$ of $q$ in $N$, and…
Yuugi
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Prove that nondegenerate zeros are isolated.

a) Prove that nondegenerate zeros are isolated. b) Furthermore, show that at a nondegenrate zero $x$, $ind_x (\vec v)=+1$ if the isomorphism $d(\vec v_x )$ preserve orientation, and $ind_x (\vec v)=-1$ if the isomorphism $d(\vec v_x )$ reserve…
XiaoXiao Zhen
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Why the map $z→z+ \overline z^m $ has fixed point with local Lefschetz number $m$ at the the origin of C (m≥0)?

My professor went through an example in class and making following claim The map $z→z+z^m$ has a fixed point with local Letfschetz number $m$ at the origin of $C$ $(m>0)$ For any $c≠0$, the homotopic map $z→z+z^m+c$ is Lefschetz with $m$ fixed…
XiaoXiao Zhen
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Preimage of 0 for a differentiable function.

If a subset $N$ of a manifold $M$ can be written as $f^{-1}(\{0\})$ being $f:M \longrightarrow \mathbb{R}$ a differentiable function, can I conclude that $N$ is a submanifold of $M$?
uuuuu
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How to prove that a certain set is a submanifold.

Let $P^{n-1}(\mathbb{R})$ the real projective space of dimension $n-1$. Consider the set $$B=\{(x,y)\in\mathbb{R}^n \times P^{n-1}(\mathbb{R}) / x=(x_1,..,x_N), y=[y_1;..;y_N], x_iy_j=x_jy_i \hspace{0.2cm} \forall i,j =1-n\}$$ The problem is to show…
uuuuu
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Prove that the Möbius band is not orientable.

Prove that the Möbius band is not orientable. I know that in the Möbius band the central circle is orientable. If I let $Y$ be the Möbius band and $Z$ be a compact submanifold of $Y$ with $\dim(Z)=\frac 12 \dim(Y)$. From one of my exercise I have…
Diane Vanderwaif
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Question on the proof of the Poincare-Hopf theorem

OP is reading Milnor's celebrated Topology from the Differentiable Viewpoint's 6th chapter, where he deals with indices of vector fields and in particular the Poincare-Hopf theorem. A lemma he used is stated and proved as following: For any…
Zhipu 'Wilson' Zhao
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every $f \in F^k_p$ has a Taylor expansion

$F_p$ is the set of germs of functions on a manifold M which vanish at $p \in M$. Let $F^k_p$ be the ideal of $C^\infty(p)$ generated by $f_1,... \,f_k$, where $f_i \in F_p$. (i.e $F^k_p$ is $\sum g_if_{i1}...f_{ik}, g_i \in \mathbb C^\infty(p),…
lll
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Prove existence of trajectory on $\mathbb{R}^2$

This question is asked on my differential topology mock mid-term, but I can't figure out what to do: Consider smooth curves $\gamma_i: \mathbb{R} \to \mathbb{R}^2, i = 1, . . . , n$ which represent the trajectories of $n$ moving obstacles (this…
Krijn
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Submersion preserves openness

Can you help me with this, but please don't post solutions, just give hints :) $M, N$ are manifolds, $f : M → N$ is a submersion, and $U \subset M$ is open, then $f(U)$ is open in $N$.
user181662
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Apply the theorem of tubular neighborhood

$M$ is a connected manifold, $N\subset M$ is a connected submanifold with nontrivial normal bundle, and dimM-dimN=1. How to prove $M-N$ is connected? There is a hint to use the tubular neighborhood theorem. Any suggestion will be grateful!
Kira Yamato
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