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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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Show that at a zero $x$, the derivative $dv_x:T_x(X)\rightarrow \mathbb{R}^N$ actually carries $T_x(X)$ into itself.

A vector field $v$ on a manifold $X$ in $\mathbb{R}^N$ is a particular type of map $v:X \rightarrow \mathbb{R}^N$. Show that at a zero $x$, the derivative $dv_x:T_x(X)\rightarrow \mathbb{R}^N$ actually carries $T_x(X)$ into itself. There is…
Extremal
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Prove that 0 is a regular value of g composed with f

I'm stuck with the following proof. Suppose $X$ and $Y$ are smooth manifolds, and $f : X \rightarrow Y$ is smooth. Let $Z$ be a smooth submanifold of $Y$. Suppose that $c:=\text{codim}_YZ$ and that $g:Y \rightarrow {\rm I\!R}^c$ is smooth and that…
bphi
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Problem on Independent Functions

Show that the curve $t\rightarrow (t,t^2,t^3)$ embeds $\mathbb{R}$ into $\mathbb{R^3}$. Find two independent functions that globally define the image. Are your functions independent on all of $\mathbb{R^3}$ or just an open neighborhood of the…
Extremal
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Prove image of derivative equal to set of smooth functions

I'm trying to prove $$ \text{the image of }d_{p}f = \big\{\alpha'(t_{0}) \; \big\vert\; \text{for all smooth }\alpha:((t_{0}-\epsilon), (t_{0}+\epsilon)) \rightarrow M \text{ such that } \alpha(t_{0})= f(p) \big\} $$ For $\quad f:U\rightarrow M$ So…
bphi
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Exists plane to intersect $M$ transversally

We have a smooth embedded submanifold $M\subseteq\mathbb{R}^n$ and we want to show that there is a $c\in \mathbb{R}$ such that the plane $x_1=c$ intersects $M$ transversally. Show that the set of $c$ is not necessarily open in $\mathbb{R}$. I have…
Cameron
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How do i calculate the rotation of a vector field only using the definition of rotation, without any coordinates?

Given any vector field , how do i either: calculate the rotation using vector calculus definition, use the differential forms definition of rotation and use differential geometry to show at leat that the vector field is integrable( has zero…
Tyson
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Show that $S^n \times R$ is parallelizable for all n

A manifold $M^n$ is called parallelizable if and only if there exist n vector fields on $M^n$ which are independent at each point of $M^n$. Could you help me figure out the vector fields for $S^n \times R$?
Yeezus
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Show $T_p M$ is isomorphic to $D_p(M)$.

Here is the problem: Let $M$ be a smooth manifold and $p\in M$ a given point. Let $D_p(M)$ be the set of derivations of $C^{\infty}(M)$ at $p$ meaning that $D_p(M)$ is the set of linear maps $\zeta:C^{\infty}(M)\rightarrow \mathbb{R}$ satisfying…
Ruth
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A smooth topological embedding that is not a smooth embedding

I am reading Introduction to smooth manifolds by Lee. In example 4.18 it says that the map $\gamma: \mathbb{R} \rightarrow \mathbb{R}^2$ given by $\gamma(t) = (t^3,0)$ is not a smooth embedding, since $\gamma '(0)= 0$. I am confused why the fact…
user110320
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Regular value of $g \circ f$ is a regular value of $g$

Given smooth maps $f: X \to Y, g: Y \to Z$, where $X, Y, Z$ are boundaryless, compact manifolds of dimension $n$, is the statement in the title true?
JJJ
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$RP^1$ is not a regular level surface of any $C^1$ map from $RP^2$ into $R$

Since $RP^2$ is compact and connected, its continuous image in $R$ is a closed interval. Let $f$ be this map. Suppose $RP^1 = f^{-1}(c)$. If $c$ is in the interior of $[a,b]$ then $RP^2$ \ $RP^1$ is not connected. Hence $c=a$ or $c=b$. WLOG…
TerryL
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Merging two zeroes of a vector field on $\Bbb R^n$

Suppose we have a vector field v on $\Bbb R^n$ with exactly two isolated zeros (call them $p,q$) which are connected by a flow-line of the vector field. Furthermore, assume one can modify the vector field in a compact nbhd of the flow line and…
Luigi M
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Almost All Hyperplanes are Not Tangent

I am trying to solve this problem: Let $M$ be an $n$-dimensional manifold embedded in $\mathbb{R}^{n + 1}$. Then almost every hyperplane in $\mathbb{R}^{n + 1}$ is not tangent to $M$ at any point. The hint given is to consider the map $f: M \to S^n$…
J126
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Differentiable submanifold : $\mathbb{R}^n \cap \{(x_1,...,x_n) : x_1^2 + x_2^3 + ... + x_n^{n+1}=1 \}$

I'm a beginner with this notion. I begin to learn it with some examples of introduction. But the definition is not clear for me. For example, $\mathbb{R}^n \cap \{(x_1,...,x_n) : x_1^2 + x_2^3 + ... + x_n^{n+1}=1 \}$ is a differential submanifold…
Mélanie De la Cheminée
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Lift of a function f of the torus

I have defined a lift of a function $f: \mathbb{R}/\mathbb{Z} \rightarrow \mathbb{R}/\mathbb{Z}$ as $\hat{f}(t)=f_0+\int_0^t f'([s]) ds$, where $f_0$ is the lift to $\mathbb{R}$ of $f([0])$ and $f'\in\mathbb{R}$ is well defined taking the same lift…
gmirsan
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