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.

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Hartman-Grobman Theorem - Necessary?

The Hartman-Grobman theorem states, in layman's terms, that a nonlinear system and the corresponding liniarized system behave similarly around a hyperbolic equilibrium point (in terms of vector fields ...
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Do the level sets of a constant rank map give a foliation of the domain?

I'm working through Smooth Manifolds by Lee and came to problem 19-8, where you're asked to prove that the level sets of a submersion form a foliation of the domain. However when I try to solve this ...
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A new definition of a focal point

Let $X$ be a manifold lying in $\mathbb{R^n}$, with $dim(X) = n-1$. Define $h: N(X) \to \mathbb{R^n}$ by $h(x, v) = x + v$, where $$N(X) = \{(x, v) \in X \times \mathbb{R^n}: v \perp T_x(X)\}$$ is the ...
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Immersions are open maps

Let $M,N$ be manifolds with $\dim M = \dim N$. If $f:M\to N$ is an immersion then $f$ is open. I thought that I have solved it, but then I thought there could be a mistake: Let $p\in M$. As $f$ is ...
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Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?

Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$. ...
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Embedding of $2$-torus minus one point into $\mathbb{R}^2$ as an open set?

Let $M^2$ be the $2$-torus minus one point. Is there an embedding of $M^2$ into $\mathbb{R}^2$ as an open set?
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Can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$? [duplicate]

As the question title suggests, can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$, i.e. in codimension $1$?
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Immersion of $M^n$ into $\mathbb{R}^n$, is $M^n$ orientable? Compact? [closed]

Say we have an immersion of $M^n$ into $\mathbb{R}^n$ (same dimension). I have two questions. Is $M^n$ orientable? Is $M^n$ compact? Thanks in advance!
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Show the parametrized torus is a 2-dimensional smooth submanifold of$\mathbb{R}^3$ [duplicate]

How can I show that the parametrized torus $T=\{(x,y,z)\in \mathbb{R}^3 : (\sqrt{x^2 +y^2}-a)^2 +z^2 =b^2 \}$ is a 2-dimensional smooth submanifold of $\mathbb{R}^3$ ? I was thinking of using the ...
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Let $f : A\subset \mathbb{R}^{n+1} \to \mathbb{R}$, what does mean that $f$ is a submersion?

I am trying to answer the following question: Let $M_a := \{ (x^1,\ldots,x^n,x^{n+1}) \in \mathbb{R}^{n+1} : (x^1)^2 + \cdots +(x^n)^2 - (x^{n+1})^2 = a\}$. For which values of $a$, $M_a$ is a ...
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36 views

If $dF_p$ is nonsingular, then $F(p)\in$ Int$N$

Here is the problem 4-2 in John Lee's introduction to smooth manifolds: Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold with boundary, and $F: M \to N$ is ...
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22 views

Properties related to connectedness of manifold

Suppose a manifold $M$ is connected. (Here I assume that a manifold is a Hausdorff, second countable space and each point $x\in M$ has a neighbourhood homeomorphic to $\mathbb{R}^{n}$, where $n$ can ...
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Induced orientation on manifold

Suppose $M$ is a naturally oriented $n$-manifold embedded in $\mathbb{R}^n$ with non-empty boundary. Give $\partial M$ the induced orientation. The induced orientation defines a unit normal vector ...
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42 views

What does Hochschild (co)homology mean

What does hochschild (co)homology mean. Is it a statement of a topological invariant? What is a way of picturing what it is doing. I am only asking this question because my experience with other ...
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25 views

Trajectories of vector fields on compact manifolds

Suppose that $X$ is a smooth vector field on a smooth manifold $M$. The trajectories of $X$ are curves $p(t)$ in $M$ which satisfy $d{p(t)}/{dt} = X(p(t))$. It's well known that $p(t)$ exists ...
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Diffeomorphism $F((z,v), \lambda) = (z, v + \lambda z)$

Show that $F: TS^n \times \mathbb{R} \to S^n \times \mathbb{R}^{n+1}$ given by $F((z,v), \lambda) = (z, v + \lambda z)$ is a diffeomorphism. Here we interpret tangent vectors to a submanifold as being ...
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Mathematical definition of the word “generic” as in “generic” singularity or “generic” map?

I've been trying to work out what generic means but I'm not making much progress. You can find an example of the usage of the word generic for example here: "School on Generic Singularities in ...
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proper submersion

I have the following question: Let $X,W$ be smooth manifolds with $W\subset X\times \mathbb{R}\times \mathbb{R}^n$ and the projection $p_{1}:W\rightarrow X$ a surjective submersion. Let $a:X\...
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Uniqueness of a PDE solution

Suppose $F:U \to \mathbb{R}^n$ is a $\mathcal{C}^1$ vector field on an open set $U \subseteq \mathbb{R}^n$. Let $\lambda \in \mathbb{C}$. Consider the partial differential equation $$F(x) \cdot \...
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1answer
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Frobenius theorem for differential forms

I have to check the next version of the theorem Frobenius: Let $M$ a smooth manifold and $\{\omega^1,\ldots,\omega^k\}\subset\Omega^{1}(U)$ $l.i.$ on $U\subset M$ and $P(x)=\{ v\in T_x M\vert \omega^...
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Gaussian curvature of $S^3$

It is easy to see that Gauss curvature of $S^2$ is $1/R^2$. How can we find the Gaussian curvature of $S^3$? What about $S^n$ in general?
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Degree of Gauss map coincides with Euler characteristic

Let $M^n \subset \mathbb{R}^{n+1}$ be a compact hypersurface, oriented with the smooth normal vector field $N(X) \perp T_xM$. Let $G: M^n \to S^n$ be the corresponding Gauss map. Does it follow that $\...
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Diffeomorphisms between smooth manifolds with boundary

In the professor Lee's introduction to smooth manifolds 2nd edition, the notion of diffeomorphism is defined for smooth manifolds with or without boundary. However, I saw some propositions that seems ...
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1answer
38 views

Definition of Integral Morse Homology

I am reading through "Morse Homology and Floer Homology" by Audin and Damian and I am confused about the definition of the differential in integral Morse homology. Let $V$ be a compact manifold, $f:...
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Showing that $\{(x,|x|):x\in\mathbb{R}\}$ is not the image of an immersion [duplicate]

Show that $A=\{(x,|x|):x\in\mathbb{R}\}$ is not the image of an immersion $f:\mathbb{R}\to\mathbb{R}^2$. Here are the definitions I know: Let $M$ be a differentiable manifold. A tangent vector at $...
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137 views

Differentiable function under specific topological constraints

Can you give an example of$\:\:\emptyset\neq D\large⊂$$\:\mathbb{R}$ and a differentiable function $f$ : $D → \mathbb{R}$ such that $\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:D ⊂$$Acc(D)$, $f^{(1)}(...
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Isometric spheres in euclidean space

I would like to prove that the sphere of radius $R>0$, $S^2(R)\subset \mathbb{R}^3$, with the induced metric is isometric to the sphere with radius $1$, $S^2\subset \mathbb{R}^3$, furnished with ...
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Riemannian metric given in polar coordinates

the Riemannian metric of the euclidean plane is given in polar coordinates as \begin{align*} ds^2=dr^2+r^2d\theta^2. \end{align*} Consider more generally, \begin{align*} ds^2=dr^2+\psi(r)^2d\theta^2, \...
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Show there exists a unique map $g$ such that $g \circ f_{2} = h$

I was wondering if somebody could give me some help on this question. Any hints etc. would be greatly appreciated. Let $0 < b < a$. Define a smooth map $h: \mathbb{R}^2 \rightarrow \mathbb{R}^3$...
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Classification of $O(2)$-bundles in terms of characteristic classes.

It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by their first Chern class. I was wondering what was the characterization of $O(2)$-bundles in terms ...
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Find a diffeomorphism between $SO(3)$ and $\mathbb{R}P^3$

Find a diffeomorphism between $SO(3)$ and $\mathbb{R}P^3$ $SO(3) =$ {${A \in M_3(\mathbb{R}) : A^TA = I, \det(A) = 1}$}, which is the special orthogonal group. And $\mathbb{R}P^3$ is the real ...
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$H_d^k(n\text{-torus})$ isomorphic to $\Lambda^k(\mathbb{R}^n)$.

Let $M^n = \mathbb{R}^n/\mathbb{Z}^n$ be the $n$-torus. Show that $H_d^k(M^n)$ is isomorphic to $\Lambda^k(\mathbb{R}^n)$. My thoughts on the problem are as follows. The map $G_{ty}(x) = x + ty$, $...
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Differentiable Version of the Jordan-Brouwer Separation Theorem

The Jordan-Brouwer separation theorem is a celebrated result of algebraic topology, which generalizes Jordan curve Theorem (it was proved independently by Lebesgue and Brouwer in 1911: see Dieudonné, ...
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Intrinsic definition of differential k-form on smooth manifold

Suppose I have a $k$-dimensional manifold embedded in $\mathbb{R}^n$. Munkres defines a $k$-form on $M$ as a a function $\omega$ that assigns an alternating tensor at each point $p \in M$ that acts on ...
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Sphere of normals

In my situation, I've constructed a quotient space $M/\sim$ in $\mathbb{R}^d$ that must be a $2$-manifold. If $M/ \sim$ is a sphere then I know that its normal spaces must vary at least 90 degrees. ...
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Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
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Describe an atlas of smoothly related charts for the Special Orthogonal Group $SO(3)$

The Special Orthogonal Group $SO(3) =$ {${A \in M_3(\mathbb{R}) : A^TA = I, det(A) = 1}$} I have successfully shown that $SO(3)$ is a manifold, but I am having a difficult time explicitly finding a ...
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1answer
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If a surface has a differentiable Gauss map, then it has an orientation?

If a surface $S\subset R^3$ has a differentiable Gauss map $N:S\rightarrow S^2$, then $S$ has an orientation? How can I prove this statement? (Here, orientation is defined by a choice of equivalence ...
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Question on the definition of outward normal vector from Spivak, Calculus on Manifolds

The following definition of the outward unit normal at the boundary of a manifold $M \subseteq \mathbb R^n$ is taken from Spivak, Calculus on manifolds (page 119). If $M$ is a $k$-dimensional ...
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The defintion of orientation of a manifold from Spivak, Calculus on Manifolds

In Spivak Calculus on Manifolds the author uses a definition of orientation of a manifold which I do not understand, and which I do not found elsewhere. I cite: It is often necessary to choose an ...
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Prove topological space has countable basis

Given a topological subspace M of $\mathbb{R}^2 \times S^1$ defined by $(x,y,e^{i\theta})$ and two charts $(U,h)$, $(V,k)$ such that $H:\mathbb{R} \times (-\pi,\pi) \to M$ $H(x,\theta) = (x,x\tan(\...
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1answer
28 views

A particular case of Sard's Theorem

I want to prove a particular case of Sard's Theorem, obviously without using the main Sard's Theorem. Let $f:M\to N$ be a differentiable ($C^{\infty}$) function. If $m=\dim M<\dim N=n$, then $f(...
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1answer
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Question on how differential form as defined for subsets of $\mathbb R^n$ and integration on them in Spivak, Calculus on Manifolds

If $V$ is a vector space, denote by $\Lambda^k(V)$ the space of alternating multilinear maps from $V^k$ to $\mathbb R$, i.e. the space of alternating $k$-tensors. Also for a point $p \in \mathbb R^n$ ...
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If $\delta c = 0$, does it follow that $d\xi = 0$?

Let $\mathcal{U} = \{\mathcal{U}_\alpha\}_{\alpha = \infty}^\infty$ be a locally finite open covering of the manifold $M^n$, with smooth functions $\lambda_\alpha$, compactly supported in $U_\alpha$. ...
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De Rahm cohomology of a sphere, help with proof

I am working through Guillemin and Pollack's proof that the de Rahm cohomology of the sphere is $H^p(\mathbf{S}^k) = \mathbf{R}$ for $p = 0$ and $p = k$ and $H^p(\mathbf{S}^k) = 0$ otherwise. Here, $...
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1answer
51 views

If a manifold has a submanifold, then the local space is a cartesian product or splits in some other way?

The following definitions are taken from Marsden et al. Manifolds, Tensor Analysis, and Applications. Definition 1: Let $S$ be a set. A chart on $S$ is a bijection $\varphi$ from a subset $U$ of $...
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41 views

A question on an exercise to show that unit sphere could not be covered by a single chart

The following is an exercise from Marsden et al, Manifolds, Tensor Analysis, and Applications in the first chapter on manifolds. First let me cite three essential definitions: Definition 1: Let $S$...
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If $M = \partial W$, with $W$ parallelizable, then an embedding $\iota : M \to S^{n+k}$ extends to an embedding $W \to D^{n+k+1}$

Suppose $M$ is an $n$-dimensional $s$-parallelizable manifold which is the boundary of the parallelizable compact manifold $W$. It is claimed in Milnor & Kervaire's Groups of Homotopy Spheres ...
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1answer
49 views

Fibration over contractible space is homotopic to a fiber

Let $\pi: E \to B$ be a fibration of $E$ over $B$, let $F = \pi^{-1}(b)$ for some $b \in B$ be a representative fiber, and suppose that $B$ is contractible. Is it always the case (or are there some ...
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56 views

Lie bracket of exact differential one-forms

Let $(M,g)$ be a Riemannian manifold. The musical isomorphisms $^\flat:\chi(M) \to \Omega^1(M)$ and $^\sharp:\Omega^1(M) \to \chi(M)$ allow the space of differential one-forms $\Omega^1(M)$ to be ...