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

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orthonormal vector fields

In a Riemannian manifold $(M^n,g)$, is it true that given a point $p$ we can find $n$ vector fields $X_i$ on a neighborhood $U$ of $p$ such that $g(X_i,X_j)(x)=\delta_{ij}$ with $x \in U$? It seems ...
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3answers
107 views

How much does $\operatorname{Aut}(H_1(S))$ determine a homeomorphism $S \to S$?

Let $S$ be an orientable compact surface. A homeomorphism $f: S \to S$ induces an isomorphism $f_{*}: H_1(S) \to H_1(S)$. How much can we say the converse? Namely, if we are given an element of ...
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1answer
283 views

Can any smooth manifold be realized as the zero set of some polynomials?

Is any real smooth manifold diffeomorphic to a real affine algebraic variety? (I.e. is there an "algebraic" Whitney embedding theorem?) And are all possible ways of realizing a manifold $M$ as an ...
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1answer
127 views

Example of a submanifold $S\subseteq M$ that is an immersed submanifold is more than one way?

Known uniqueness results say that an embedded submanifold has a unique smooth structure making it an embedded submanifold with the subspace topology, and immersed submanifolds have a unique smooth ...
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2answers
158 views

Differentiable manifolds $\mathscr C^k$ vs. $\mathscr C^\infty$

I noticed that there exists a (in some sense) better definition of the tangent space via the dual of a certain quotient algebra which is easier to work with in some cases. This however only works for ...
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1answer
190 views

Is the configuration space a manifold? a CW complex?

The ordered configuration space of $n$ points in a topological space $X$ is defined as $F(X,n)=\{(x_1,\ldots,x_n)\in X^{n} | x_i\neq x_j \mbox{ for } i\neq j\}$ and the unordered configuration space ...
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5answers
541 views

Different definitions of a “one-form”

I started self-studying some differential geometry while using several different sources, but I'm confused about the notion of a one-form and how different places define it differently. Here are some ...
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1answer
159 views

$K$-theory of smooth manifolds: continuous vs. smooth vector bundles

Suppose I have a smooth manifold $M$, and want to consider the $K$-theory $K^0(M)$. I could define this in the usual way (by taking the Grothendieck group of the monoid of equivalence classes of ...
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2answers
106 views

frustrating experience about differential geometry

I am felling rather frustrated now, after taking a long time to study differential geometry, but with little progress... Indeed my major is mainly numerical analysis. I am studying modern geometry, ...
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1answer
99 views

Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
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1answer
106 views

Is there any way to define differentiablity without any reference to the Euclidean space?

We define metric spaces based on the properties of the real numbers $\Bbb{R}$. In the same spirit we define smooth manifolds. But there is a more general and elegant way to formulate our intuition of ...
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1answer
160 views

Topologist's Sine Curve not a regular submanifold of $\mathbb{R^2}$?

I am trying to work out the details of following example from page 101 of Tu's An Introduction to Manifolds: Example 9.3. Let $\Gamma$ be the graph of the function $f(x) = \sin(1/x)$ on the ...
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2answers
837 views

Relationship Between Differential Forms and Vector Fields

I am trying reach an understanding of precisely how the space of differential forms is related to the space of vector fields. These are the definitions that I understand and am using for these ...
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2answers
154 views

Classical tensor analysis and Tensors on Manifolds

I learned tensors the bad way (Cartesian first, then curvilinear coordinate systems assuming a Euclidean background) and realize that I am in very bad shape trying to (finally) learn tensors on ...
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2answers
218 views

Why do folded concentric circles and rectangles form a hyperbolic paraboloid?

Here is a "self-forming" origami that I made from folding concentric circles - it would also happen if I folded concentric rectangles. How can the fold shapes such a saddle-like geometry?
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2answers
225 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 $\nabla$ ...
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2answers
386 views

Lie bracket is a connection?

In Road to Reality, section 14.6 on Lie derivative Penrose writes: Now $\epsilon^2 [j,h]$ corresponds to an $O(\epsilon^2)$ gap in the ‘parallelogram’ whose initial sides are $e_j$ and $e_h$ at ...
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1answer
282 views

Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups. On a ...
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1answer
184 views

Is a continuous map between smoothable manifolds always smoothable?

Let $X$ and $Y$ be topological manifolds and $f:X\to Y$ a continuous map. Suppose $X$ and $Y$ admit a differentiable structure (at least one). My question: is it always possible to choose a ...
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1answer
263 views

Hopf Fibration in Local Coordinates

I have the following task: Consider the unit sphere $\mathbb S^3$ in $\mathbb R^4$. We know $\mathbb CP^1\simeq \mathbb S^2$ (homeomorphic). Identifying $\mathbb R^4$ with $\mathbb C^2$, we have a ...
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2answers
598 views

Question about Lee's Introduction to Topological Manifolds

From page 2 in Lee's Introduction to topological manifolds: Question 1: What does "describe parametrically" exactly mean? Is it a synonym for "global coordinate chart"? (that is, an atlas ...
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1answer
243 views

Question about $4$-manifolds and intersection forms

This is a question related to an earlier question of mine: I've been reading about topological invariants. Some of them are defined in terms of quadratic forms. My current understanding is: we can ...
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2answers
292 views

A question about the topological homogeneity of manifolds

Let $M$ be a connected topological manifold of dimension $n \geq 1$. It is well known that if $p,q \in M$ then there exists a homeomorphism $\phi : M \to M$ such that $\phi(p) = q$. This can be summed ...
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2answers
525 views

Which mapping tori are Seifert manifolds?

According to Orlik's lecture notes on Seifert manifolds (and the Wikipedia page on Seifert fiber spaces), a mapping torus over a 2-torus is a Seifert manifold if and only if it is the mapping torus of ...
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1answer
169 views

How to find the integral curves that are orbits of one-parameter groups?

Consider $\mathbb{R}^2$ with standard symplectic structure and inner product. Consider a Hamiltonian $$H=(x,y)A(x,y)^t$$ where $$A=\begin{pmatrix} \alpha & \beta \\ \beta & \delta ...
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1answer
103 views

Interpretation of generalized eigenvector in orbits

First of all, this is my fourth question about dynamical systems in a week, sorry for that. Considering a linear bidimensional dynamical (autonomous) system, the orbits can be plotted in the phase ...
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1answer
92 views

Volume of neighborhood of the curve

Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|<r\}$. How can i show ...
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1answer
63 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 ...
5
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2answers
217 views

Injectivity of a map between manifolds

I'm learning the concepts of immersions at the moment. However I'm a bit confused when they define an immersion as a function $f: X\rightarrow Y$ where $X$ and $Y$ are manifolds with dim$X <$ ...
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2answers
1k views

Orientability of projective space

Q: Show that $\mathbb {RP}^n$ is not orientable for $n$ even. First I looked at the definition for orientability for manifolds of higher degree than 2, because for surfaces I know the definition with ...
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1answer
241 views

A question about concept of pushforward

In An Introduction to Smooth manifolds by Lee is written: for any smooth vector fields V and W on a manifold $M$, let $\theta$ be the flow of $V$, and define a vector $(\mathcal{L}_v W)_p$ at each ...
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2answers
239 views

Are matrices with determinant zero a manifold?

Consider the set of matrices with determinant zero in $M_n(\mathbb R)$, where $n > 1$. Is it a manifold? In fact, is it even a topological manifold? I would suspect not; but I do not have a proof. ...
5
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1answer
147 views

Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
5
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1answer
375 views

Line integral and integration of differential forms

The definition of integral of a $k$-form $\omega$ over a parametrized manifold $ Y_\alpha $ is $ \int_{Y_\alpha}\omega = \int_A\alpha^*\omega$ where $ \alpha\colon A \to R^n $. Let $ \gamma:(a, b) ...
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1answer
219 views

The embedding of smooth manifold

I have run into a problem in my differential geometry book. Let $M$ be a smooth manifold and $F={C^\infty }(M,\mathbb R)$. Define a mapping $i:M \to {\mathbb R^F}$ by ${i_f}(x) = f(x)$ for $x \in M,f ...
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2answers
85 views

Geometry of Commuting Hermitian Matrices

I am a physicist working on a project dedicated to the quantisation of commuting matrix models. In the appropriate formalism this problem is reduced to a quantisation in a curved space -- the space of ...
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1answer
65 views

Why are there always pairwise intersections in a Heegaard splitting?

Let $M=A\cup B$ be a Heegaard splitting, such that $\{\alpha_i\}_{i=1}^g$ is a set of boundaries for meridian disks of $A$, and $\{\beta_i\}_{i=1}^g$ is a set of boundaries for meridian disks of $B$ ...
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1answer
115 views

Submanifold given by an open immersion

I was wondering if the following is true: Let $M,N$ be two manifolds such that $\dim M\leq \dim N$ and $f:M\rightarrow N$ an smooth immersion. Assume that for any open set $U\subset M$, ...
5
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1answer
119 views

One-form on quotient manifold

Let $M$ be a smooth manifold with tangent bundle $TM$ and cotangent bundle $TM^*$ and $\psi\in TM^*$ a one-form. We denote the quotient manifold of $M$ by the free and proper $G$-action $\varphi$ as ...
5
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1answer
363 views

proof of stokes theorem

I don't understand the "idea" of the following proof, aswell as some of the steps. As i'm not sure about its "ways", im not editing it much and as such it might be in the wrong order. My sincere ...
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1answer
99 views

Embedding manifolds of constant curvature in manifolds of other curvatures

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space ...
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1answer
184 views

Finding an algebra of smooth functions on a manifold with a given product.

I am having trouble with the following exercise from Jet Nestruev's Smooth Manifolds and Observables. It is one of the first exercises in the book and I have no idea how to approach this type of ...
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1answer
298 views

Algebraist's definition of the tangent space of a manifold

By the "algebraist's definition" of the tangent space of manifolds, can we say that the partial derivative $d/dx$ belongs to the the tangent space of $S^1$? It feels strange, but I can't see why it ...
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1answer
149 views

Vector field on riemannian manifold

Let $M$ be a riemannian manifold and $N$ be a submanifold of $M$. Let $v$ be a vector field in $N$. Then $v$ can be covariantly differentiated along $\gamma$ resulting in new field $u$. (Here consider ...
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1answer
65 views

The graph of $x\mapsto |x|$ cannot be the image of an immersion.

How can one prove that the set $\{(x,|x|)\in \mathbb{R}^2 \mid x\in \mathbb{R}\}$ cannot be the image of an immersion of a smooth manifold? This was my homework exercise in a course about ...
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1answer
93 views

Slice of a coordinate system in a manifold

In the book - Foundations of differentiable manifolds and Lie groups by Frank Warner, the definition of a slice is as under. Suppose that $(U,\phi)$ is a coordinate system on $M$ (dimension $d$) with ...
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1answer
149 views

Directional, differential and lie derivatives on manifolds intuition?

Trying to translate elementary multivariable calculus into the language of manifolds: Is the directional derivative on a manifold just a way of finding the rate of change of a vector in a single ...
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1answer
131 views

Pullback of a form using the Hopf fibration

I embed the 2-sphere and the 3-sphere in $\mathbb{R}^3$ and $\mathbb{R}^4$ respectively. Then denote by $\{x_1,x_2,x_3,x_4\}$ the coordinates on the 3-sphere and $\{y_1,y_2,y_3\}$ on the 2-sphere. So ...
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1answer
302 views

Constant Rank theorem for domain with nonempty boundary

Problem 4-3 in J.M. Lee's introductory text about smooth manifolds, asks to formulate and prove a version of the constant rank theorem for a map of constant rank whose domain is a smooth manifold with ...
5
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1answer
170 views

Isometries from Diffeomorphisms

Given a compact, connected Lie group G of diffeomorphisms on a manifold M, how to construct a Riemannian metric on M such that elements of G are isometries of M?