1
vote
2answers
46 views

Some questions about the proof of the General Linear Group being a manifold.

I understand the idea behind proving that GL(n,$\mathbb{R}$) is a smooth manifold by first using the fact that it is isomorphic to $\mathbb{R}^{n^{2}}$ and using the continuity of the determinant ...
1
vote
2answers
55 views

Wedge product and determinants

I am attempting to self-study differential forms this summer, but I ran into this definition for the wedge product in my book, and it doesn't make any sense to me. Note that $\Bbb R^3_p$ is the ...
0
votes
1answer
33 views

Parametric equations of manifold

I have am working for a linear algebra test and I realized that I don't know how to solve exercises with linear manifolds even the basic one. W : $ x+y-z+u=1 $ $ 2x+u=2 $ $ z -u=0 $ I don't ...
2
votes
2answers
138 views

Grassmannian, Plucker coordinates

In which books can I find something about the grassmannian and the plucker coordinates ?
1
vote
0answers
53 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 ...
1
vote
0answers
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 ...
1
vote
1answer
41 views

Existence of a nonzero vector to form

Let $ f: \mathbb {R}^m\times \mathbb {R}^m \rightarrow \mathbb {R}^m $ an alternate form of grade two. If $ m $ is odd, prove that there exists $ v\neq 0 $ such that $ f (u, v) = 0 $, for all $ u \in ...
1
vote
1answer
36 views

Kernel of matrix with identity as submatrix

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^k$ be a $C^\infty$ map and let $X=\text{graph}f$, i.e. $$X=\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^k\mid y=f(x)\}.$$ What is the tangent space to $X$ at ...
0
votes
0answers
28 views

Express the change of coordinate matrix in terms of partial derivative of the transition map $\phi$ Where $\tilde{\sigma} =\sigma \circ \phi$

Let $\sigma :U \to W\cap S$ and $\tilde{\sigma}: \tilde{U} \to \tilde W \cap S$ Be two surface patches around $p\in W\cap \tilde W$ $T_p(S) $ be tangent plane of S at p. $T_p(S)=span\{\sigma_u, ...
3
votes
1answer
67 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 ...
4
votes
2answers
214 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 ...
1
vote
2answers
84 views

What is the geometric meaning of the number of independent derivatives of $\gamma$?

Let $\gamma:I \to \mathbb{R}^n$ be a curve. I want to see, what is a geometric meaning of the number of independent derivatives of $\gamma$. I guessed it is it's dimension but it was not. Can you help ...
0
votes
1answer
85 views

Invertibility of a function

S is a surface in $\mathbb{R}^{3}$ parameterized by a function $f:S\rightarrow(a,b)^{2}\subset\mathbb{R}^{2}$ $F$ is the function defined by: $F:T^{1}S\rightarrow(a,b)^{2}\times S^{1}$ ($T^{1}S$ is ...
3
votes
1answer
77 views

Polar decompostion should be a diffeomorphism, right?

I seem to have gotten stuck in the mud verifying what I thought was going to be a completely straightforward fact. I would appreciate if somebody could help dig me out. Inside the $n \times n$ ...
2
votes
2answers
83 views

Smooth maps on a manifold lie group

$$ \operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\ \begin{align} &n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\ &n = 2, \operatorname{GL}_n(\mathbb ...
3
votes
0answers
330 views

Real projective space is Hausdorff

I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix?? This prove is correct or I need to add something ?? ...
2
votes
1answer
81 views

Real Projective Space

Corollary $\bf7.15.$ The real projective space $\mathbb{R}P^n$ is second countable. How I can prove this corollary 7.15.I consider that ı can apply the corollary 7.10.But I could not can you help ...
1
vote
1answer
42 views

Locally finite or not

I am tryıng to learn locally finite and can you give an explanation for my green writing please thank you
3
votes
1answer
93 views

What is overlop

I want to ask something that what is overlop.My teacher said that For Ex1, Everything is overlop hence it is not locally finite.For example 2,it doesnt overlop.Actually ı could not understand.please ...
0
votes
2answers
125 views

Topological manifold example

$\theta(x,x^2)=x$ $\Bbb X =${$(x,x^2)| x$ in $\Bbb R$} And V is subset of $\Bbb R$ $dim\Bbb X=1$ My instructor said that this is topological manifold. Why? Please can you explain me? This ...
2
votes
2answers
97 views

An open cover that is not locally finite

I could not understand that why is not locally finite for example 13.4 can you give me explanation please.
2
votes
0answers
149 views

Show that the projection map is Orientation preserving iff n is even

My question is that Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere $U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on ...
1
vote
1answer
125 views

I did all explanation. Can you just teach me how to calculate this interior product?

Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Show that an orientation form on $S^n$ is $w=\sum _{i=1}^{n+1}(-1)^{i-1}x^i dx^1∧...∧dx^i∧...∧dx^{n+1}$ I ...
2
votes
1answer
94 views

Manifolds with boundary and definition

Can you help for understanding this definitions in a good way.What is the my problem is that I can not image this definition and proposition in my mind and also ı dont understand the reason that ı ...
2
votes
0answers
46 views

Orientation-preserving diffeomorphism [duplicate]

Can you help for solving this please. Although I study this subject I could not solve this question please help me ı am willing to learn this question.
4
votes
1answer
163 views

The open Möbius Band is not orientable

Can you explain my green underlying please.I have confused and ı dont understand why by the continuity of the orientation at the points $(0,0)$ and $(1,0)$ are also $e_{1},e_{2}$
4
votes
2answers
174 views

Orientations on Manifold

This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example ...
2
votes
1answer
155 views

Why is the cylinder surface on $\Bbb R^3$ orientable?

Why is the cylinder surface on $\Bbb R^3$ orientable? Please can someone explain me clearly?
1
vote
1answer
369 views

Orientation preserving diffeomorphism.

I am stuck with the question. I guess that I need to write jacobian matrix. But I could not do. Please help me thank you
1
vote
1answer
166 views

Show that the zero set of $f$ is an orientable submanifold of $\Bbb R^{n+1}$.

Suppose $f(x_1,...,x_{n+1})$ is a$ C^∞$ function on $\Bbb R^{n+1}$ with $0$ as a regular value. Show that the zero set of $f$ is an orientable submanifold of $\Bbb R_{n+1}$. In particular, the unit ...
1
vote
1answer
212 views

How to decide whether F is orientation-preserving or orientation-reversing as a diffeomorphism onto its image.

Let $U$ be the open set $(0,∞)×(0,2π)$ in the $(r,θ)$ -plane $R^2$. We define $F : U ⊂ R^2 → R^2$ by $F (r, θ ) = (r cos θ , r sin θ )$. How to decide whether F is orientation-preserving or ...
0
votes
1answer
45 views

How do I show that $F^{∗}(dx∧dy∧dz) = ρ^{2} \sin φ dρ∧dφ∧dθ$.

I dont know how to solve. Please help me. I need to understand such types of the question for my exam studyings.
1
vote
2answers
92 views

Transition formula for 1-forms

$\bf{17.3.}$ Transition formula for $1$-forms Suppose that $(U,x^1,\ldots,x^n)$ and $(V,y^1,\ldots,y^n)$ are two charts on $M$ with nonempty overlap $U\cap V$. Then a $C^\infty \;1$-form $\omega$ ...
0
votes
1answer
117 views

Problem about differential of a linear map

Please can you tell how to solve this problem clearly? Please solve this explanatorily. Thank you
1
vote
1answer
41 views

Is $VV^T + D$ a submanifold?

If the positive definite matrix P forms a manifold, is that the subset that {P: P = V V^T + D} where V is a low rank matrix and D is a positive definite matrix a sub-manifold? This idea is ...
7
votes
2answers
119 views

Finding a subspace whose intersections with other subpaces are trivial.

On p.24 of the John M. Lee's Introduction to Smooth Manifolds (2nd ed.), he constructs the smooth structure of the Grassmannian. And when he tries to show Hausdorff condition, he says that for any 2 ...
2
votes
0answers
33 views

Exponential Families and Riemannian Symmetric Spaces

Suppose the $f_{X}(x|\theta)$ is a probability density function from an exponential family. Is it true that the Riemannian manifold which has the Fisher information as it's Riemannian metric is a ...
0
votes
1answer
92 views

Prove that a surface of revolution is a 2dimension manifold

I have a question about surface of revolution. Prove that a surface of revolution is a 2dimension manifold.
1
vote
1answer
39 views

Pfaffian system of equations

Given a distribution $L^r=\{X_1,...X_r\}$, that is $\{X_1(p),...X_r(p)\}$ linearly independent on each point $p$ on a manifold $M$ of dimension $n$. At a point $p \in M$, the system ...
0
votes
2answers
78 views

Implicit function theorem : statement and the rank of the matrix

This is perhaps something standard in linear algebra(a subject I am somewhat weak in). So I apologize in advance. I would be grateful if someone can guide me. The statement of implicit function ...
2
votes
1answer
97 views

Generalization of Grassmann manifold to include translations?

I came across a certain generalization of Grassmann manifolds and was wondering what work if any has been done on it. If you take the space of $n\times p$ real matrices, $n>p$, and define an ...
1
vote
2answers
149 views

The dimension of linear map

I am reading "Introduction to smooth manifolds" by Lee and one place is very unclear for me: Let $P$ and $Q$ be any complementary subspaces of $V$ (which is an $n$-dimensional real vector space) of ...
1
vote
2answers
48 views

Find $A^{-1}$(W) of linear manifold W

Given linear map $A:\mathbb{R}^2\to \mathbb{R}^4$ defined as $$A = \begin{pmatrix} 1 & 1 \\ 1 & -1 \\ 0 & 2 \\ 3 & 1 \end{pmatrix}$$ and linear manifold $ W \subset ...
1
vote
1answer
191 views

Dimension of the space of matrices with constant determinant.

I'm looking for the dimension of the space of $n\times n$ real matrices $A$ such that $\det(A)=c$. I apply 2 different approaches and I get different answers. which one is correct? 1) So we ...
3
votes
1answer
177 views

Dimension of the tangent space

This question is regarding the dimension of the tangent space $T_p\mathbb{R}^n$ as it is defined in the context of smooth manifolds. One the one hand, $T_p \mathbb{R}^n$ can just be interpreted as a ...
0
votes
1answer
49 views

Maximum dimensionality of the linear manifold

What is the maximum dimensionality of the linear manifold spanned by: ...