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

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54 views

Few questions about global analysis relating $C^k$ functions

First question is about the definition. Let $U$ be an open subset of $R^n$. Let $f$ be $k$ times continuously differentiable function on $U$. $C^k$ norm of $f$ is defined as sum of uniform norm of ...
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41 views

Question related to tangent space of $U(n)$ at a matrix $g\in U(n)$

I was working on a homework problem that involved showing that the map $f:U(n)\rightarrow S^1,g\mapsto det(g)$ is a submersion (which is given here) And the following question emerged: Given $g\in ...
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2answers
124 views

Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

Is this statement true: Question: Can any N dimensional manifold be realized as a boundary of some N+1 dimensional manifold? If so/not, how to prove/disprove it? I read a TQFT paper from Edward ...
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56 views

Nonvanishing vector fields on $\mathbf{S}^2$ and bases for $T_p\mathbf{S}^2$.

A vector field on a manifold is a (continuous, differentiable) map $X: M \to TM$ such that $X(p) \in T_pM$ for each $p \in M$. The tangent space $T_pM$ has a basis. Take $M = \mathbf{S}^2$. For each ...
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42 views

Showing that a map $h:S^2\rightarrow \mathbb{R}^4$ is an immersion

The Problem Let $h:S^2\rightarrow \mathbb{R}^4$ be a smooth map of the form $$ h(x,y,z)=(zy,yz,zx,ax^2+by^2).$$ Show that $h$ is an immersion for any $a,b\in \mathbb{R},a,b\neq 0,ab<0$. Attempt ...
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101 views

Unitary group and unit circle

Let $U(n)$ denote the group of complex unitary matrices, let $S^1$ be the unit circle in the complex plane. Then the map $$f:U(n)\to S^1,\quad f(A)=det(A)$$ is a group homomorphism and a submersion. ...
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86 views

Related to tangent space of points on the sphere

Let $S^n$ be the $n-$dimensional unit sphere. Define $h: S^3\to S^2$ and $F:S^1\times S^3\to S^3$ as follows $$h(x)=(2(x_1x_3+x_2x_4),2(x_2x_3-x_1x_4),(x_1^2+x_2^2)-(x_3^2+x_4^2))$$ ...
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2answers
51 views

Hessian of Morse function on $S^{n}$ mistake

I am trying to get that $f(x_{0},...,x_{n+1})=x_{n+1}$ has $index_{(0,...,0,1)}=n$ Can you find my mistake or post a partial solution? My attempt I evaluate df using inverse of stereographic ...
3
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1answer
168 views

Use of cut-off functions and partitions of unity

This is a simple problem, but I would still be very thankful if you could give me an advice on it. I'm trying to show that in a compact M-dimensional manifold, $$\int e^w \sqrt{g}\, dx \leq C \exp ...
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51 views

Finding a diffeomorphism between two smooth structures of $\Bbb R$

This is taking from Tu's Introduction to Manifolds book. We have defined $\mathbb{R}$ as the real line with the differentiable structure given by the maximal atlas of the chart ...
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2answers
94 views

Is this set a manifold?

For which $ ( \alpha , \beta ) \in \Bbb R^2$ set: $\{ (x_1,x_2,x_3,x_4) \in \Bbb R^4 | x_1+x_4= \alpha, x_1 x_4 - x_2x_3 = \beta \}$ is a manifold? I made a Jacobian matrix: $ \begin{bmatrix} ...
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1answer
61 views

Torus, manifolds

I have some trouble with the following questions: $\mathbb{R}^3$ has standard coördinates $(x, y, z)$. Regard in the plane $x=0$ the circle with centre $(x,y,z) = (0,0,b)$ and radius $a$, ...
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2answers
57 views

How to motivate vectors as derivations?

In a manifold it's easy to motivate the definition of vectors as equivalence classes of curves. On the other hand the definition as derivations is harder to motivate. I know how to show that the space ...
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1answer
58 views

Show that a set is a manifold.

Let $n \ge 3 $. How can I show that $M:= \{(x_1,...,x_n) \in \Bbb R^n \setminus \{(0,...,0)\} | x_1^2+...+x_n^2 = x_1 \cdot...\cdot x_n \}$ is a manifold of class $C^1$? Can anyone please tell me ...
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192 views

Munkres' Question on Manifolds

In Munkres' 'Analysis on Manifolds' on pg. 208 there's a question which reads: QUESTION: Let $f:\mathbb R^{n+k}\to \mathbb R^n$ be of class $\mathscr C^r$. Let $M$ be the set of all the points ...
2
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1answer
39 views

Intersection of two open sets in the projective plane

I want to compute the cohomology groups of the real projective plane, $P^2$, using Mayer Vietoris exact sequence. Now $H^0(P^2)=\mathbb{R}$, $H^2(P^2)=0$ being $P$ not orientable, so my problem really ...
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1answer
237 views

Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have ...
2
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1answer
52 views

vector field on $\mathbb{R} P^2$

Actually this is a quesion in Lee's book, Manifolds and differential geometry. I have problems working with projective spaces as manifolds.(e.g. what are curves in projective spaces ? I need to know ...
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1answer
64 views

Orientation of $X \times Y$

Suppose that $X$ is not orientable. How can I show that $X \times Y$ is never orientable, no matter what manifold $Y$ may be? I've tried supposing that $X \times Y$ is orientable, then using that ...
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1answer
56 views

Degree of polynomial seen as a smooth map

I need some help with a part of an exercise. Let $P$ be a real polynomial of degree $d$, seen as a map $P:\mathbb{R}\rightarrow\mathbb{R}$. Prove that if $d$ is even then the degree of $P$, $degP$, ...
3
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2answers
152 views

If a smooth map between manifolds is injective, is the induced map on the tangent spaces injective too?

If $\phi:M\longrightarrow N$ is an injective smooth map between two manifolds, then is $d\phi_m:M_m\longrightarrow N_{\phi(m)}$, the induced map between the tangent spaces injective too? I tried the ...
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1answer
36 views

Extension of funcion

I think its right but Im not sure. I have topological space (exactly manifold - second countable, Hausdorff, local Euclidean topological space) M, dim M=m. Let $A \subset M$ is closed set, dim A=n, ...
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1answer
38 views

Which integral curves of a field are defined for all times t?

Which integral curves of the field $X=x^2 \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}$ are defined for all times t? I would be very thankful if somebody can help me understand what ...
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52 views

Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
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1answer
29 views

Relative Compactness $\Rightarrow$ Compactness

I try to figure out: $(\overline{A}^U\text{ compact in }U )\Rightarrow( \overline{A}^X\text{ compact in }X)$ ...while $U\in\mathcal{T}$ It's clear for the case: $\overline{A}^X\subseteq U$ But else, ...
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1answer
90 views

Question about statement of Rank Theorem in Rudin

Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ mapping of an open set $E\subset \mathbb{R}^n$ into $\mathbb{R}^m$, and $F'(x)$ has rank $r$ for every $x\in E$. ...
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2answers
79 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...
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27 views

About a function space on $\bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}$

For each $t$, let $\Gamma(t)$ be a $C^k$ hypersurface without boundary. Define $$Q = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$ I am trying to understand some properties of this space $Q$. ...
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34 views

Don't understand an integration by parts result involving a step function on spacetime domain

I'm reading this work. Let $\Omega$ be a bounded (open) domain, and define $Q=(0,T)\times\Omega$. For every $t \in [0,T]$, let $\Omega_1(t), \Omega_2(t)$ be open subsets of $\Omega$, with $S(t)$ the ...
4
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1answer
86 views

How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?

I am reading this paper. Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$ On page 5 of ...
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1answer
19 views

About the boundary of a set of the form $Q_i = \bigcup_{t \in (0,T)}\Omega_i(t) \times \{t\}$

Let $\Omega$ be a bounded (open) domain. For every $t \in [0,T]$, let $\Omega_1(t), \Omega_2(t)$ be open subsets of $\Omega$, with $S(t)$ the interface separating $\Omega_1(t)$ and $\Omega_2(t)$. ...
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1answer
39 views

Curve in $\mathbb{P}^{n}(\mathbb{R})$, differentiable manifolds

I need a book which contains the demonstration that if $\Gamma$ is a curve in $\mathbb{P}^{n}(\mathbb{R})$ defined as $F(x_{1}, \cdots \ \ , x_{n+1}) = 0 $ , with $F$ homogeneous polynomial, then ...
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1answer
19 views

$C^k$ hypersurfaces can be split in this way?

Let $S$ be a bounded $C^k$ hypersurface of dimension $n \geq 2$ in $\mathbb{R}^{n+1}$ with no boundary. Is it true that $S$ can be split into two hypersurfaces $S_1, S_2$ that have boundary, and a ...
1
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1answer
35 views

Tangent space, a group and a manifold

Let G be a group with a smooth manifold structure and let $u:G\times G\rightarrow G$ be the smooth multiplication defined by $(x, y)\mapsto xy$. Question: Why is the Tangent map $T_{(e, e)}u$ given ...
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1answer
76 views

Orientability of Grassmannians

How can I understand when $Gr(n,k)$ is orientable and when not? I found that answer is yes if and only if $n \vdots 2$, but I do not know how to prove it.
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32 views

uniqueness of Hopf invariant

(Hopf invariant, page 427 of A. Hatcher's Algebrac Topology): Let $f: S^m\longrightarrow S^n$ with $m\geq n$. We can form a CW-complex $C_f$ by attaching a cell $e^{m+1}$ to $S^n$ via $f$. The ...
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1answer
155 views

What does it mean “being geodesic” is not invariant?

We know that "being geodesic" is not invariant under re-parametrization. Only affine re-parametrization preserves the property of being a geodesic. Also, a geodesic is locally distance minimizer. ...
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46 views

Chains and cochains: integer versus real coefficients

Let a real, smooth manifold $M$ be given. For each non-negative integer $k$, let a singular $k$-cube on $M$ be a continuous mapping $c:[0,1]\to M$. Let $C_k(M,\mathbb Z)$ denote the set of formal ...
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31 views

Do we need to pay attention to the codomain of a differentiable function?

I came across the following definitions: We call $M\subset \mathbb R^N$ $m$-dimensional $C^k$-submanifold of $\mathbb R^N$ if for all $a\in M$ there is an open neighborhood $U$ of $0$ in $\mathbb ...
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25 views

Prime Manifold in Different Dimensions

Definition Let $M$ be an $n$ dimensional manifold. If $M$ = $M_1$ # $M_2$, we have $M_1=\mathbb S^n$ or $M_2=\mathbb S^n$. Then $M$ is called a prime manifold. $\mathbb T^2$ and $\mathbb RP^2$ ...
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19 views

Prime Decomposition in Two Dimensional Manifold

There is Prime Decomposition in three dimensional manifold, so I want to ask is there Prime Decomposition in two dimensional manifold. I think no because $\mathbb T^2$#$\mathbb RP^2$=$\mathbb ...
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1answer
36 views

Dependence on the class of differentiability between manifolds and maps

Maybe a silly question, but in some books (like "Differential Geometry - Manifolds, Curves, Surfaces - Gostiaux and Berger"), when differentiable maps of class $C^s$ are defined, we have something ...
3
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1answer
86 views

Integral curves of $X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}$ on a cylinder

Consider coordinates $(\theta, z)$ on $S^1 \times \mathbb R$, and a vector field $$X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}.$$ Show that the integral curve of ...
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2answers
136 views

de Rham cohomology of $\mathbb R^2 \setminus \mathbb Z^2 $

I am trying to calculate the cohomology of $X = \mathbb R^2 \setminus \lbrace \mathbb Z \times \mathbb Z \rbrace = \lbrace (x,y) \in \mathbb R^2 : x \text{ and } y \not \in \mathbb Z \rbrace.$ ...
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41 views

Definition of Manifold's Orientation

I am reading the book of manifold. And I find there are many definitions about one object, such as orientation, Euler character and degree of map. I am confused with the conception of orientation. ...
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49 views

Tanget space to manifold via curves without map

I define tangent space T to differentiable manifold, in point p, via equivalence class of curves. The condition for this equivalence is $(\varphi\circ\gamma_1)'(0)=(\varphi\circ\gamma_2)'(0)$ for some ...
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53 views

$\mathbb{RP}^2$ does not embed into $\mathbb{R}^3$: reduction to the differentiable case

It is not difficult to see that the real projective plane cannot be embedded into $\mathbb{R}^3$ as a differentiable submanifold (for example one can easily show that the complement would consist of ...
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2answers
79 views

A subset $A$ of a manifold $X$ that is a manifold but not a submanifold of $X$

Let $X$ be a manifold and $A$ a subset of $X$. Is it possible for $A$ to be a manifold without being a submanifold of $X$. Thank you for your help!!
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27 views

Proving a global property of the connection from a property of local connection matrices

Let $D$ be an $m$-dimensional distribution on an $n$-dimensional manifold $M$. Let $U\subset M$ be an arbitrary open subset such that on $U$ we can define vector fields $e_i$ such that for each $x\in ...
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40 views

Tensor vector bundle construction

$\newcommand{\p}{\partial}$Let $M$ be a smooth manifold, and define $$T_{r,s} := \bigsqcup_{p \in M} (T_p M)_{r,s} = \bigsqcup_{p \in M} \big( \underbrace{T_p M \otimes \dots \otimes T_p M}_r \otimes ...