For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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Is $S^0$ a manifold?

Consider a singleton space $\{x\}$, it is a manifold and it is locally euclidean as there is a homeomorphism to $\mathbb{R}^0$. However, consider $S^0=\{-1,1\}$ with the discrete topology, there does ...
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2answers
511 views

Product of manifolds & orientability

I'm studying orientability of manifolds currently and I'm having trouble to prove the following: $M\times N$ is orientable iff $M$ and $N$ are orientable. I am able to prove that the product is ...
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58 views

how to obtain a generalized Morse function out of a fiber bundle?

Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth ...
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63 views

Triangulation of a 3-sphere

If one wants to generate a Simplicial complex of the topology of the 3-sphere, one can just take the boundary of a 5-cell, 16-cell or 600-cell. The curvature is concentrated on the edges meeting the ...
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1answer
54 views

Proving orientability of manifold

I don't know how to prove the following: $RP^n$ is orientable manifold if n is odd? Any help is welcome.
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1answer
79 views

A Milnor Differential Topology Excercise

If $m<p$, show that every map $f:M^m\longrightarrow\ S^p$ is homotopic to a constant, where $M^m$ is smooth manifold of dimension $m$. I tried to show that $M^m$ is contractible or convex, but I ...
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1answer
75 views

Distinction between a vector and a tensor of type (1,0)

Let's say I have a differentiable manifold $\mathscr{M}$. A vector $v$ on this manifold is a map from $\mathscr{F}$ to $\mathbb{R}$, where $\mathscr{F}$ is the set of all smooth functions from ...
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1answer
118 views

Bump function has a compact support?

Sorry for the basic question, but couldn't find the answer. We say that the bump function $\phi(x)=e^{-1/(1-x^2)}$ has a compact support. However, $\phi(x)\neq 0$ only for $x \in [0,1)$, which means ...
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91 views

Orientation manifold, what is wrong with my argument?

As I learned, a manifold M is oriented if there exists a smooth nowhere-vanishing n-form on M. So, I am very doubting about the following construction of a n-form $\omega$ on any smooth manifold M (M ...
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1answer
47 views

Moving a compact submanifold off of another submanifold?

This is an intuitive idea that I see referenced a lot. Consider the following situation. Let $M$ and $N$ be submanifolds, $M$ compact, in some larger manifold $X$. Suppose also that ...
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1answer
32 views

Tangent vectors as curves equivalence relation

I do not understand the definition of the equivalence relation that is defined on the curves creating a tangent vector space. Let $X$ be any manifold, a point $x \in X$, two curves $\alpha:(-a,a) \to ...
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2answers
39 views

Can I identify $S_k(V)$ with an homogeneous space?

I'm in trouble with a question: Let $V$ be an $n$-dimensional vector space over $\mathbb R$. Can I identify the manifold, $$S_k(V):=\{(X_1, \ldots, X_k): X_1, \ldots, X_k\in V\ \textrm{are linearly ...
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1answer
67 views

Proof of 'manifold with dimension less then 4 always has differentiable structure'

L.S., I read in the lecture notes of my course on manifolds (undergraduate) a little side-note that stated that every manifold with dimension less then 4 can be equipped with a differentiable ...
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1answer
173 views

Differentiable manifolds, uniqueness of maximal atlases and definition of smooth manifolds maps.

I have proved that given an atlas for a topological space $M$ that a maximal atlas containing $M$ is unique. But my proof would fail to generalise to the statement that a maximal atlas conatining a ...
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0answers
75 views

Constructing vector bundles from local covers and transitions functions

Let $M$ be a smooth manifold. Suppose we are given an open cover ${U_\alpha}$ of $M$ and for $\alpha,\beta$ ; a smooth map $\tau_{\alpha\beta}\colon U_\alpha \cap U_\beta \to GL(k; R)$ satisfying the ...
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1answer
58 views

Tangent spheres to a differentiable manifold

I have the following problem. Let $M$ be a compact $C^r$-manifold (with $r>1$) of dimension $m$ embedded in an euclidean space of dimension $k$. I am told that then there is some $\epsilon >0$ ...
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1answer
96 views

Indices Contraction in Minkowski Spacetime

Why is it that $$\partial_\mu\partial^\mu=\partial_t^2-\nabla^2$$ (this I believe is called the D'Alembert operator.) but $$\partial_\mu j^\mu=\dot{j^0}+\nabla\cdot \vec j?$$ Why is there a minus on ...
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1answer
62 views

Immersion and Embedding problem

Prove that: A $C^r$-immersion $f:M \rightarrow N$ is a local $C^r$-embedding. A differential map $f: \mathbb{R}^n \rightarrow \mathbb{R}^m, f \in C^r, r\geq 0, m\geq n$ is a immersion if ...
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1answer
73 views

Real projective $n$ space

We define $\sim$ on $\mathbf{R}^n - \{0\}$ by $x \sim y$ if $x = \lambda y$ for some $\lambda \in \mathbf{R}$. We define projective $n$ space by $X = (\mathbf{R}^n - \{0\})/{\sim}$. I am having ...
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27 views

Doubt on proof showing matrices of certain rank form a submanifold. [duplicate]

I have gone through two proofs to show that matrices of rank $k$, where $0 \leq k \leq \min(m,n)$ form a submanifold of the set $ M(m \times n ,\mathbb{R}) $. Both proofs involve writing down an ...
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1answer
62 views

Prove that $g^{-1}(0)$ is a $n$-dimensional manifold.

Let $A\subset \mathbb R ^n$ be open and let $g:A\to \mathbb R ^p$ be a differentiable function such that $g'(x)$ has rank $p$ whenever $g(x)=0$. Then $g^{-1}(0)$ is an $(n-p)$-dimensional manifold. ...
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1answer
73 views

Why do we consider tangent spaces and what not, when we can just use Whiteney's Embedding Theorem and do calculus in $\mathbb{R^{2m}}$?

Given a real $m$-dimensional smooth manifold why do we consider tangent spaces and what not, when we can just use Whiteney's Embedding Theorem and do calculus in $\mathbb{R^{2m}}$? I assume there is ...
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64 views

Equivalent definitions of Tangent space - 2

L.S., In my book Vector Analysis by Klaus Jänich, Three different 'versions' of the Tangent space of a point $p$ at a differentiable variety are being discussed. The 'geometrical': the set of ...
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1answer
145 views

Equivalent definitions of the tangent space

L.S., In my book Vector Analysis by Klaus Jänich, Three different 'versions' of the Tangent space of a point $p$ at a differentiable variety are being discussed. The 'geometrical': the set of ...
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1answer
142 views

$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$

Show that for a curve lying on a sphere of radius r with nowhere vanishing torsion, the following equation is satisfied: $$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$$ Please ...
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2answers
453 views

The tangent space of a manifold at a point given as the kernel of the jacobian of a submersion

Let $\phi:M\to N$ is a smooth map, $q\in N$ a regular value, and $V=\phi^{-1}(q)$. I want to show that, for each $p\in V$, $T_p(V)= \mathrm{ker}(\phi_*)\subseteq T_p(M)$ (where $\phi_*$ is the ...
4
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1answer
61 views

Show that $\dot{n_s}=-\kappa_s t$

I found the question in a differential geometry textbook while studying. This question seems so intesting to me. So please help me solving it. Show that, if $\gamma$ is a unit-speed plane curve, ...
2
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1answer
86 views

Verify that an ellipse has four vertices.

Verify that an ellipse has four vertices. The ellipse is given by $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ And I took $$x=a\cos t$$ and $$y=b \sin t$$ for $t\in [0,2\pi]$ Please can someone help ...
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569 views

Distinguishing the Cylinder from a “full-twist” Möbius strip

Playing around with the definition of a fiber bundle, I found that while a Möbius strip (with its usual "half-twist") is a nontrivial fiber bundle, it seems that a Möbius strip with a "full-twist" is ...
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2answers
177 views

How do manifolds have enough structure to do calculus?

I am referring, of course, to to differentiable manifolds. I've seen a few different definitions. The one I like best is the one which says it's a topological space such that every point has a ...
2
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1answer
114 views

derivative on manifold

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth function and $M$ be a smooth manifold of R^n. Assume that $Df(x)v \neq 0$ for all $v$ being tangent to $M$ at $x$ and for all $x$ in $M$. Can we say ...
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1answer
61 views

injective function on manifold

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth function and $M$ be a smooth manifold of R^n. Assume that $Df(x)v \neq 0$ for all $v$ being tangent to $M$ at $x$ and for all $x$ in $M$. Can we say ...
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135 views

Submanifold of a regular value of a manifold with boundary

Question: Suppose $M$ is a smooth manifold with boundary, $N$ is a smooth manifold, and $F:M\rightarrow N$ is a smooth map. Let $S=F^{-1}(c)$, where $c\in N$ is a regular value of both $F$ and ...
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2answers
100 views

How to calculate Frenet-Serret equations

How to calculate Frenet-Serret equations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =\left(\cos \left(\frac{s}{\sqrt 2}\right), \sin \left(\frac{s}{\sqrt 2}\right), ...
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1answer
72 views

Show that TpM is a vector space of dimension n.

Show that TpM is an n-dimensional vector space. Hint: Given two tangent vectors v1 and v2 at p with corresponding curves γ1 and γ2, we can “add” the corresponding curves in Rn and then move back to M: ...
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83 views

the equivalence between paracompactness and second countablity in a locally Euclidean and $T_2$ space

suppose $M$ is a locally Euclidean Hausdorff space, show that $M$ is second countable if and only if it is paracompact and has countable components. This is Problem 2-15 in Introduction to smooth ...
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1answer
2k views

how to calculate the curvature of an ellipse

how can I compute the curvative of an ellipse given by $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ do i need to take $x=acos(t)$ and $y=bsin(t)$? please show me a way how to solve this? thank you for ...
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1answer
227 views

To Prove That a Certain Set is a Manifold

Definitions and Notation: Let us write $\underbrace{\mathbb R^n\times \cdots\times\mathbb R^n}_{m \text{ times}}$ as $(\mathbb R^n)^m$. A rigid motion in $\mathbb R^n$ is a function $L:\mathbb ...
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2answers
52 views

the differential of a function $f\in C^{\infty}(M)$: two definitions

Let $M$ be a smooth $n$-dimensional manifold (on $\mathbb R$). If $p\in M$, we have that $\Big\{\frac{\partial}{\partial x^1}\Big|_p,\ldots,\frac{\partial}{\partial x^n}\Big|_p\Big\}$ is a basis for ...
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1answer
73 views

example of a topological space such that there exists a sequence that escapes to infinity but has convergent subsequence

Find an example of a topological space such that there exists a sequence that escapes to infinity but has a convergent subsequence This actually is from exercise 2.15 of introduction to smooth ...
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151 views

homomorphisms of $C^{\infty}(\mathbb R^{n})$

Let $F: \mathbb R^{n} \to \mathbb R^{m} $ be a smooth map, then we have homomorphism of algebras $F^{*}: C^{\infty}(\mathbb R^{m}) \to C^{\infty}(\mathbb R^{n})$. Is it true that any homomorphism of ...
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1answer
120 views

How to show that the limaçon has only two vertices.

Question: Show that the limaçon has only two vertices. I researched what is limaçon. And I reached the following result; Note that I only know that The limaçon is the parametrized curve ...
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1answer
201 views

Writing a parametrization of the cissoid by using $\theta$

The cissoid of Diocles is the curve whose equation in terms of polar coordinates $(r,\theta)$ is $$r = \sin\theta \tan\theta, −\pi/2 < \theta < \theta/2$$ Write down a parametrization of the ...
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242 views

surface vs differentiable manifold

Every surface is a smooth manifold, but the reciprocal is verified? some concrete example of a differentiable manifold is not surface? Thanks in advance for the suggestions.
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3answers
104 views

What dimensions are possible for contours of smooth non-constant $\mathbb R^n\to\mathbb R$ functions?

While for $n=2$ it is pretty clear that the contours of a non-constant $f:\mathbb R^n\to\mathbb R$ are either extrema (and therefore points) or (the union of) 1-dimensional isolines, for $n=3$ I am ...
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1answer
45 views

Locally Finite Cover

Give a hint to the following problem: Let $M$ be a second-countable manifold, $N\subset M$ closed subset, $\Omega\supset N$ its open neighbourhood. Then $M$ has an locally finite cover $\{U_i\}_{i\in ...
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102 views

Why isn't the graph of $y=|x|$ a smooth manifold? [duplicate]

Consider the graph of $y=|x|$ from $-1<x<1$. Equip it with a single chart, the projection onto the $x$-axis. Is it now a smooth manifold? It seems like it shouldn't be smooth, but perhaps with ...
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0answers
48 views

Manifolds - Characterization of Graph

I'm trying to prove the following: Suppose $M$ and $N$ are smooth manifolds and $S \subset M \times N$ is an immersed submanifold. Let $\pi$ denote the projection $M \times N \to M$. TFAE: (i) ...
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4answers
531 views

What is the codimension of matrices of rank $r$ as a manifold?

I'm reading through G&P's Differential Topology book, but I hit a wall at the end of section 4. There is a result stating The set $X=\{A\in M_{m\times n}(\mathbb{R}):\mathrm{rk}(A)=r\}$ is a ...
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2answers
316 views

Compatibility of a connection and metric

Every Riemannian manifold admits a metric connection. Suppose $M $ is a manifold and $\nabla $ is an arbitrary connection on the tangent bundle. Does $M$ necessarily admit a metric such that $\nabla$ ...