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

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3
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2answers
123 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 ...
3
votes
2answers
139 views

Show that the map on spheres is smooth

For each of the following maps between spheres, compute sufficiently many coordinate representations to prove that it is smooth. $(a):$ $p_{n}:\mathbb{S}^{1}\rightarrow \mathbb{S}^{1}$ is the $n$th ...
0
votes
2answers
73 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!!
0
votes
1answer
43 views

Intersection of a manifold with open set

I'm using Flemming's book Function of Several Variables. In it, the author defines Manifolds like this: Let $1\le r\lt n,\ q\ge1$. A nonempty set $M \subset \mathbb{R}^n$ is a manifold of dimension ...
2
votes
2answers
45 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 ...
0
votes
0answers
51 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 $$h(x)=(2(x_1x_3+x_2x_4),2(x_2x_3-x_1x_4),(x_1^2+x_2^2)-(x_3^2+x_4^2))$$ ...
0
votes
2answers
40 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 ...
1
vote
2answers
53 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 ...
3
votes
2answers
87 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} ...
1
vote
1answer
53 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 $a$, ...
2
votes
1answer
582 views

An open subset of a manifold is a manifold

Let $M$ be an $n$-manifold. I would like to show that any open subset $A$ of $M$ is an $n$ manifold. Let $x\in A$. since $M$ is a manifold, there exists a neighborhood $U_x\subseteq M$ of $x$ such ...
1
vote
1answer
52 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 ...
12
votes
3answers
841 views

concrete examples of tangent bundles of smooth manifolds for standard spaces

I'm having trouble visualizing what the topology/atlas of a tangent bundle $TM$ looks like, for a smooth manifold $M$. I know that $$\dim(TM)=2\dim(M).$$ Do the tangent bundles of the following ...
21
votes
1answer
354 views

How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
1
vote
2answers
77 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 ...
2
votes
1answer
36 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 ...
6
votes
1answer
104 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
votes
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 ...
6
votes
1answer
60 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 ...
1
vote
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, ...
3
votes
1answer
43 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 $P$, $degP$, ...
2
votes
1answer
30 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 ...
1
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0answers
40 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 ...
0
votes
1answer
63 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 $x\in E$. ...
4
votes
1answer
77 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 ...
0
votes
0answers
26 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 $Q$. ...
3
votes
2answers
126 views

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

I am trying to calculate the cohomology of $X = \mathbb R^2 \setminus \lbrace \mathbb Z \times \mathbb Z \rbrace = \lbrace (x,y) \in \mathbb R^2 : x \text{ and } y \not \in \mathbb Z \rbrace.$ ...
1
vote
0answers
25 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 ...
0
votes
1answer
18 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 $\Omega_2(t)$. ...
0
votes
1answer
37 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 ...
1
vote
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
vote
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 ...
4
votes
1answer
113 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. ...
2
votes
1answer
38 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.
2
votes
0answers
22 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 ...
3
votes
1answer
74 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 ...
1
vote
0answers
42 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 ...
1
vote
2answers
42 views

What is the equivalent of Delaunay tringulations in high dimensions?

For 2D manifolds, Delaunay triangulation is a very useful tool for coarse graining. It has the nice property that in the flat/euclidian manifold case, it reduces to a 2D simplicial tesselation of the ...
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0answers
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 ...
2
votes
0answers
23 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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0answers
18 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 ...
1
vote
1answer
32 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 ...
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0answers
33 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. ...
1
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0answers
47 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 ...
2
votes
0answers
47 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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0answers
25 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 ...
1
vote
0answers
36 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 ...
0
votes
0answers
68 views

Every element of the tangent space of a manifold $M$ is the tangent to a smooth curve in $M$.

I am reading the first chapter on Manifolds from the book Warner, in which I have the following doubt. Let $M$ be a differentiable manifold. A $C^{\infty}$ mapping $\sigma:(a,b)\longrightarrow M$ is ...
1
vote
1answer
26 views

Involutive Properties of Space-structures on Smooth Manifolds

I am currently reading Quantum Invariants of Knots and 3-Manifolds by Turaev, and I am having a hard time understanding a statement made on page 120. He is explaining the property of space-structures, ...
6
votes
1answer
88 views

Geodesics of one-dimensional manifold

I apologize if my post is "silly" because I don't know much about riemannian geometry. I know that $M = (0,1)$ (the open unit interval) can be seen as a one-dimensional manifold. Since $M$ is an ...