Tagged Questions
2
votes
2answers
51 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 ...
2
votes
0answers
51 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 ?? ...
1
vote
1answer
53 views
Real Projective Space
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 me.
1
vote
1answer
31 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
73 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
67 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 ...
1
vote
2answers
55 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
74 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
109 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
53 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 ...
1
vote
0answers
26 views
Trivialization of a path of tamed almost complex structures
I am wondering if the following result is true:
Let $(V,\omega)$ be a symplectic vector space and $\{J_t\}_{0\leq t\leq 1}$ a smooth path of almost complex structures on $V$ which are tamed by ...
2
votes
0answers
36 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
72 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
84 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
81 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?
0
votes
1answer
73 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
votes
0answers
25 views
Imaginary line passing through non-collinear points in R3.
I have come to a problem where n points are provided in 3-Dimensional plane. I need a imaginary line which can be assumed that it is passing through these points.
1
vote
1answer
63 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 ...
0
votes
1answer
33 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
41 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.
3
votes
2answers
125 views
Relationship between trace of a linear map and the number of points it fixes.
Problem Statement: Let $\Phi_A:T^2\rightarrow T^2$ be a smooth mapping into the torus induced by a linear map $A\in SL_2(\mathbb{Z})$ under the quotient relation that identifies 0 and 1. Assume that A ...
1
vote
1answer
38 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 ...
0
votes
0answers
38 views
How is the shape operator related to the second fundamental form?
I don't understand how the shape operator $\Theta $ is related to the second fundamental form $\Pi$.
Before now, we have derived the second fundamental form as a quadratic form with the matrix ...
0
votes
1answer
47 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.
4
votes
1answer
73 views
Alternative Almost Complex Structures
Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure gives $V$ the structure of a complex vector ...
1
vote
0answers
47 views
Why does the map $x^2$ have constant rank?
I'm just trying to wrap my head around the rank of a map via some examples. Now, if I have the smooth map of manifolds $F:\mathbb{R} \to \mathbb{R}, F(p) = p^2$, then the differential is given by ...
1
vote
1answer
37 views
Vector space structure on $(-1,1) \subset \mathbb{R}$ (or: möbius strip as vector bundle)
I'm first putting the question into it's context, so probably you can see if i'm asking the wrong question to get what i want.
The Task is to show that the Möbius (Moebius) strip is a Vector bundle ...
3
votes
1answer
99 views
Two Definitions of the Special Orthogonal Lie Algebra
I am encountering two definitions of the special orthogonal lie algebra, and I would like to know if they are equivalent, and if there are advantages to working with one over the other.
If we begin ...
3
votes
1answer
39 views
Find point $X$ such that line through plane $E$ and sphere $S$ meet at $(0,0,1)$ (stereographic projection)
Find the point $X$ such that the line going through the plane $E$ and sphere $S$ meet at the point $(0,0,1)$ (stereographic projection).
Let $S$ denote the unit sphere
$$S = \{(x,y,z) \in ...
2
votes
1answer
63 views
Stereographic Projection: Find the point $X$ where the plane $E$ meets the sphere $S$
Let $S$ denote the unit sphere
$$S = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 = 1\}$$
and $E$ denote the plane in $\mathbb{R}^3$ given by $z = 0$
$$E = \{(x,y,z) \in ...
3
votes
1answer
63 views
Pullbacks and transpose map
Given maifolds $M,N$ and a smooth map $\phi:M \to N$, and a smooth function $f:N \to \mathbb{R}$, we have the pullback of $\phi$ by $f$ to be the function $\phi^* f = f \circ \phi : M \to \mathbb{R}$. ...
3
votes
3answers
135 views
Fit a quadratic form given covariant derivatives on the sphere?
I am trying to solve for a particular vector given covariant first and second derivative for a function on a sphere. If you have a quadratic form restricted to the sphere:
$f(x) = ...
2
votes
1answer
59 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
0answers
48 views
About decomposition of three-forms
Patrick D Baier in his PhD thesis in chapter2 in page 14 for proving the theorem 2.1.4 used of following non-trivial fact
Let $0\neq X\in V $(here $V$ is of dimension 6) , $W^*=Ann(X)$ and ...
3
votes
1answer
102 views
Outward vectors to an Ellipsoid and Euclidean metrics
I'm reading Arnold's proof of the topologically equivalence of the equations $\dot{x}=Ax$ and $\dot{x}=x$ when all the eigenvalues of the $n \times n$-matrix $A$ have positive real part. The proof is ...
2
votes
1answer
73 views
Projective Plane with three points
I am reading some texts on projective geometry but I am still confused about some easy exercises. I found the following one:
$P_1=[0:1:2:3], P_2=[0:1:2:4], P_3=[1:1:1:1]$ are three points in $\mathbb ...
1
vote
0answers
67 views
Symmetry of the Ricci tensor of the first kind
I am looking to show that the Ricci tensor of the first kind, $R_{i j}$ obtained by retracting the Riemann tensor of the first kind, via $R_{i j} = R^k_{i j k}$ is symmetry. To do this, I have shown ...
1
vote
1answer
117 views
Diffeomorphism on the torus
Let $S : \mathbb{R}^n → \mathbb{R}^n$ be linear invertible map, then $S$ projects to $\mathbb{T}^n$ diffeomorphism
if and only if $S ∈ GL_n(\mathbb{Z})$.
I can't prove the right to left implication.
3
votes
0answers
81 views
Curvature of particular Riemannian metric
Let $U = \{ (x_1, \dots, x_n) \mid x_j > 0 \text{ for all } j\}$ and let $\|x\|^2 = \sum_j x_j^2$. The function $x \mapsto -\log \|x\|^2$ is strictly convex on $U$ and thus defines a Riemannian ...
1
vote
0answers
53 views
General definition of a line
In the book on Linear Algebra that I am using, the author defines a line in an arbitrary vector space $V$, given direction $ 0 \neq d \in V $ and passing through $ p \in V$ as
$ l(p;d)= \lbrace v\in ...
1
vote
0answers
89 views
Submanifold of $\mathbb{R}^4$
In the space of $2\times 2$ matrices, find explicitly the sets of matrices with 1)a single zero eigenvalue, 2) a pair of pure imaginary eigenvalues. Show that each set is a submanifold of ...
4
votes
1answer
217 views
Non surjectivity of the exponential map to GL(2,R)
I was asked to show that the exponential map $\exp: \mathfrak{g} \mapsto G$ is not surjective by proving that the matrix $\left(\matrix{-1 & 0 \\ 0 & -2}\right)\in \text{GL}(2,\mathbb{R})$ ...
2
votes
1answer
333 views
How to evaluate the derivatives of matrix inverse?
Cliff Taubes wrote in his differential geometry book that:
We now calculate the directional derivatives of the map $$M\rightarrow M^{-1}$$ Let $\alpha\in M(n,\mathbb{R})$ denote any given matrix. ...
4
votes
0answers
210 views
Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?
Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem".
I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
1
vote
1answer
122 views
There is a closed non-zero $n$-form on $\text{GL}(n, \Bbb{R})$
How to Prove that
There is a closed non-zero $n$-form $\omega$ on $\text{GL}(n, \mathbb{R})$ which is left and right invariant.
0
votes
1answer
71 views
What are the allowed dimensions for vector fields?
When one first encounters the concept of vector field, especially in physics, it is often presented just as n-tuple of numbers $(x_1, x_2, \ldots , x_n)$ prescribed to each point. In this manner $n$ ...
2
votes
2answers
219 views
Openness of $\varphi(U_Q \cap U_{Q'})$ in the definition of Grassmannian Manifolds (Lee: Introduction to Smooth Manifolds)
I am reading Lee's Introduction to Smooth Manifolds and I have some problems with definition of Grassmannian manifold given in Example 1.24, p.22. I'll write the details below.
My question is:
Why ...
0
votes
2answers
85 views
Tensored vectorspaces isomorphic to the endomorphisms [duplicate]
Possible Duplicate:
Understanding isomorphic equivalences of tensor product
I have the following question: Let $V$ be a vectorspace with an inner product $<.,.>$. Let $V^{*}$ be its ...
0
votes
1answer
123 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 ...
2
votes
3answers
216 views
Proposition about curves in $S^2$
Let $\gamma_1,\gamma_2:(a,b)\to S^2$ be unit speed curves in $S^2=\{\vec{v}\in\mathbb{R^3}:\vec{v}\cdot\vec{v}=1\}$. Then the following two statements are equivalent:
(1) There is a $3\times 3$ ...
