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

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42 views

Jacobi fields and variations

I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a ...
5
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1answer
334 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 ...
3
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1answer
65 views

Explaining the definition of vector bundles

Recall the definition of a vector bundle: Let $M$ be a topological space. A $k$-dimensional vector bundle over $M$ is a topological space $E$ with a surjective continuous map $\pi\colon E \to ...
3
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1answer
34 views

Definition of submanifolds by regular values

Let $f: M \rightarrow N$ and $q \in N$ be a regular value, then $f^{-1}(q)$ is a submanifold of $M$. Now assume that $q \in N$ is not a regular value, but you pick $K:=f^{-1}(q) \cap \{p \in M; ...
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1answer
37 views

Unit radial vector field

Lee's book defines the unit radial vector field in normal coordinates as $$ \partial_r:= \frac{x^i}{r(x)} \partial_i$$ and $r(x):=\sqrt{\sum_i (x^i)^2}$ Now this is a unit vector field iff ...
5
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2answers
53 views

Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$

PROBLEM: Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$, where $M$ is a connected smooth manifold and $p,q \in M$ , $X_p \in T_pM$ and $Y_q \in T_qM$ I know ...
8
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1answer
158 views

Translating a passage of a paper by L. Bérard Bergery

I am currently studying the following paper on Einstein manifolds: L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d'Einstein, Inst. Elie Cartan, Univ. Nancy №6, 1-60 (1983). I have ...
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0answers
32 views

Preimage of a small normal deformation under an embedding again a normal defomation?

Let $j\colon M\rightarrow W$ be a smooth embedding of smooth manifolds and assume $M$ and $N$ have Riemannian metrics, but $j$ is not necessarily an isometric embedding. Let $N\subseteq W$ be a ...
3
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0answers
73 views

Differentiating an endomorphism

Let $(M,\rho)$ be a symplectic $2$-dimensional manifold, and let $J$ be a compatible complex structure on $M$, i.e. the symmetric $(0,2)$-tensor $$g(*,*) = \rho(*,J*)$$ is a Riemannian metric. Denote ...
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56 views

Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
3
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1answer
35 views

Homology and Neighborhood

Let $X$ a connected manifold, $x \in X$ and $V$ a neighborhood of $x$. Assume $i:V \to X$ induce isomorphism between all homology groups. Does $X-p$ and $V-p$ still have the same homology groups ? ...
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25 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
2
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1answer
57 views

Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
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1answer
54 views

Injectivity of the Differential of Smooth Map

I am trying to answer the following question: Let $M = \{(x,y)\in \mathbf{R}^2 : x^2 + y^2 < 1\}$. Define a smooth or $C^\infty$ function by $f\colon M \rightarrow \mathbf{R}^2$ as ...
2
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1answer
36 views

Compute in the chosen charts of $M$ and $S^1$ the expression of $DF_{(5,0,-4)}$

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5 x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. We ...
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0answers
32 views

Let $p=(5,0,-4)$ and $v \in T_{(5,0,-4)}M$. Compute $(F^{*}\omega)_p(v)$.

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. ...
4
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1answer
55 views

Parametrizing the time an element stays in an open subset

Let $X$ be a topological space (If it helps anything, we can assume $X\subseteq\mathbb{R}^n$ or $X$ being a smooth manifold.) and $U\subseteq [0,1]\times X$ an open subset. Does there exist a ...
2
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1answer
25 views

What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...
3
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1answer
51 views

Tangent space as derivations exercise

Thinking of the tangent space to a manifold as derivations is a concept which just kind of eludes me. I am comfortable thinking about tangent vectors as equivalence classes of curves and with the ...
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1answer
28 views

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth?

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth? I can't find a formal definition. I know, that we say, that $\partial\Omega$ has a ...
0
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1answer
23 views

$M\subset\mathbb{R}^n$ is a open subset, $p\in M$ is arbitrary. Find $T_pM$ and $N_pM$

PROBLEM: $M\subset\mathbb{R}^n$ is a open subset, $p\in M$ is arbitrary. Find $T_pM$ and $N_pM$. I know how to determine $T_pM$ and $N_pM$ for explicit examples, but I dont know how to handle this ...
3
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1answer
30 views

Existence of a vector field which dominates the first local vector fields given by the charts of a locally finite covering

Let $M$ be a smooth manifold, let $\{U_i,\psi_i\}_{i\in I}$ be locally finite family of charts and let $K_i\subseteq U_i$ be compact subsets. Does there exist a vector field $X$ on $M$, such that ...
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1answer
47 views

Open product neighbourhoods

Let $X\times Y$ be a topological space (if it helps anything, it can be assumed to be a smooth manifold and $U\subseteq X\times Y$ an open subset. Let $x\in X$ and $O\subseteq Y$ be open, such that ...
8
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1answer
1k views

Isometries preserve geodesics

Let $f$ be an isometry (i.e a diffeomorphism which preserves the Riemannian metrics) between Riemannian manifolds $(M,g)$ and $(N,h).$ One can argue that $f$ also preserves the induced metrics $d_1, ...
4
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1answer
74 views

Group action and smooth manifolds

I was wondering if it is for a compact (i.e. Hausdorff) smooth manifold $M$ sufficient to have a free group action of a finite group $G$ in order to conclude that $M/G$ is a compact smooth manifold? ...
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0answers
77 views

Check Riemannian manifold's isometry to $\Bbb{R}^n$

Let $\mathcal{M}$ be the convex cone of symmetric positive definite $n\times n$ real matrices. $\mathcal{M}$ is an $\frac{n(n+1)}{2}$-dimenasional Riemannian manifold. Could you help me proving (or ...
2
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0answers
24 views

How to show that the inverse image of a face of a simple polytope is a connected manifold?

If $M^{2n}$ is a toric manifold over a simple polytope $P^n$ i.e; the orbit space of the action of the $(S^1)^n$ on $M^{2n}$ is an $n$ dimensional simple polytope $P^n$. Let $\pi : M^{2n} \rightarrow ...
3
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0answers
66 views

$L^\infty$-bounds on eigenfunctions of Laplace-Beltrami opeator

Let $w_k$ be the eigenfunctions of the Laplace-Beltrami operator on a compact manifold $M$ without boundary. We assume that $\{w_k\}$ are orthonormal, thus $\|w_k \|_{L^2} = 1$. We know $w_k$ are ...
2
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1answer
65 views

Is this one infinite dimensional manifold?

First of all, just to give context to the question: I've been reading some articles in Physics, and those articles imply without proof that one space is one infinite dimensional manifold. One of those ...
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0answers
13 views

Two vector fields are cojugate but not take orbits

Let X and Y be C1 vector feilds on R^m. Suppose that 0 is an attracting hyperbolic singularity for X and Y. Show that there exists a homemorphism h of a neighborhood of origin which conjugate the ...
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0answers
15 views

Quickest way to restrict a homeomorphism

Let $\phi: U \to V \subset \mathbb{R}^n$ homeomorphism. My desire is: I want to say the restriction $\phi|_{\phi^{-1}(B_{r'}(x))}:\phi^{-1}(B_{r'}(x)) \to B_{r'}(x) $ is a homeomorphism in the ...
2
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3answers
43 views

2-Sphere surface coordinate dimension

Ordinary sphere in $\mathbb{R}^3$ is two-dimensional object (2-sphere), i.e. it requires at least two coordinates to define point on a surface. As I notice, however, there is a catch. If we use ...
0
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1answer
53 views

Smooth function, which separates between a closed and a open set.

Let $M$ be a smooth manifold, $O\subseteq M$ an open subset and $B\subseteq M$ a closed subset, such that $closure(O)\subseteq interior(B)$ I think there must exist a smooth function $f\colon ...
1
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1answer
39 views

Homeotopy of Shrinking Manifolds

Let $M$ be a $n$-dimensional open manifold in $\mathbb{R}^n$. Let $B^n_k$ be the closed $n$-dimensional ball of radius $k$. Let $$N_k = (M^c \oplus B^n_k)^c$$ where $X^c$ denotes the complement of ...
6
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1answer
157 views

Definition of a Cartesian coordinate system

Apologies if this is a basic question, but I'd really like to clarify the exact meaning of what a Cartesian coordinate system is. Heuristically, is it correct to say that a Cartesian coordinate system ...
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0answers
55 views

Equivalence of two norms on $L^p(M)$, $M$ compact manifold.

Let $(M,g)$ be a compact Riemannian manifold, $\mu(g)$ the Riemannian Lebesgue measure. Then we can define the usual $L^p$-spaces (lets assume $p<\infty$), $L^p(M,g):=L^p(M,\mu(g))$. For $f\in ...
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2answers
119 views

Why the lens space L(2,1) is homeomorphic to $\mathbb{R}P^3$?

According to one definition of lens space $L(p,q)$, which is gluing two solid tori with a map $h:T^2_1 \rightarrow T^2_2$. And $h(m_1)=pl_2+qm_2$, $l_i$ means longitude and $m_i$ means meridian of the ...
2
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1answer
37 views

Integration of $V$-valued differential form

When studying fibre bundles, connections and gauge theories it is usual to consider vector-valued differential forms, like the connection one-form, or it's pull back by a local trivialization known as ...
5
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1answer
95 views

Theorema egregium violated in dimension $n \ge 4$?

Gauß showed that for surfaces in $\mathbb{R}^3$ the Gaussian curvature ( = sectional curvature) is invariant under local isometries. This is known as the thema egregium. Now in another question ...
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1answer
37 views

(Locally) sym., homogenous spaces and space forms

We had some definitions of particular types of Riemannian manifolds in our lecture 1.) Locally symmetric spaces. They were Riemannian manifolds with the property that $\nabla R=0$ everywhere. 2.) ...
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1answer
67 views

Meaning of “locally homeomorphic to $\mathbb{R}^{n}$”

I am fairly new to differential geometry and approaching it with a physics background (in the study of general relativity), as a result I'm having a few struggles with terminology etc, so please bear ...
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0answers
13 views

Are variables in embedded space Statistically independent variables?

Performing Taken's phase space delay embedding on the observations $\mathbf{z}$ of a univariate random variable, with an embedding dimension $d$, we get a realization of $n$ points such as: ...
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0answers
49 views

Constructing coordinate maps on manifolds

I've been studying differential geometry for a little while now, but I've never properly justified to myself rigorously the need to consider other more general coordinate maps, other than Cartesian on ...
2
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1answer
67 views

Centroid of manifold

The centroid of a compact manifold is defined by the following equation: $c(Y_a)$ is the centroid of the parametrized manifold $Y_a$ is the point in $\Bbb R^n$ whose $i^{th}$ coordinate is given by ...
3
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3answers
38 views

Inverse function of $f(x,y,z) = (xy-z^2, x+z)$?

How do you determine the inverse function $f^{-1}: \mathbb{R}^2 \to \mathbb{R}^3$ of $f: \mathbb{R}^3 \to \mathbb{R}^2 , f(x,y,z) = (xy-z^2, x+z) $ ? Or to put it into a bigger context: ...
31
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3answers
434 views

Does every bijection of $\mathbb{Z}^2$ extend to a homeomorphism of $\mathbb{R}^2$?

Given a bijection $f\colon \mathbb{Z}^2 \to \mathbb{Z}^2$, does there always exist a homeomorphism $h\colon\mathbb{R}^2\to\mathbb{R}^2$ that agrees with $f$ on $\mathbb{Z}^2$? I don't see any ...
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0answers
15 views

characeterization of zero sets of the riemannian measure of a riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold (does not have to be orientiable). Then there exists the Riemannian measure $\nu(g)$ on $M$. Let $(U_i,x_i)$ be a finite covering of $M$ of charts and let ...
6
votes
3answers
77 views

The set of all matrix with rank $n-1$ is a hypersurface.

Prove that the set $M$ of $n\times n$ matrices with rank $n-1$ is a hypersurface in $\mathbb{R}^{n²}$ and find the tangent space at $A=(a_{ij})$ where $a_{ij}=\begin{cases} \delta_{ij} \ \text{if} ...
2
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2answers
58 views

Prove that the antipodal mapping is an isometry on $S^n$. Help understanding the proof.

Prove that the antipodal mapping $A: S^n \to S^n$ given by $A(p)=-p$ is an isometry. I know that in order to prove that a map $f$ is an isometry of a smooth manifold $M$ it must hold true that ...
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0answers
15 views

How to determine the pdf for a model in phase space representation?

Consider a univariate discrete linear model : $z(k) = y(k) -(a* z(k-1) + b * z(k-2))$ where $y(k) = x(k) + \eta(k)$ $x(k) = s(k) + p*s(k-1) + q*s(k-2)$ is a Moving Average model of order 2. ...