2
votes
1answer
48 views

Geometric Product

I have a problem with the geometric product: In my book the unit trivector is defined like this: $(e_{1}e_{2})e_{3}=e_{1}e_{2}e_{3}$ But that would mean $(e_{1}e_{2})e_{3}= (e_{1} \wedge e_{2})\cdot ...
4
votes
1answer
96 views

Changing local coordinates on a manifold by a diffeomorphism

This is the set up of my problem: Let $M$ be a manifold with local coordinates $x^1,\dots, x^n$. Let $x^1,\dots,x^n,\xi_1,\dots,\xi_n$ denote the induced coordinates on $T^\ast M$. Let $f:M\to M$ be ...
0
votes
2answers
36 views

Given a measurable vector field, construct another such that together they form a basis at every point

Let $v_1:(0,1)\rightarrow \mathbb{R}^2$ a measurable function such that $v_1(x)\neq 0$ for all $x$. I wonder if it is possible to construct a measurable function $v_2:(0,1)\rightarrow \mathbb{R}^2$ ...
4
votes
3answers
94 views

Interior product between differential forms and vector fields

I don't understand what is meant when someone writes that forms (or form fields) "eat" vectors (or vector fields). For example when I have a one form field ω=3dx+5dy+3xdz and a vector field ...
4
votes
0answers
86 views

Consistency of different ways to define the tensor product

It seems my trouble with understanding tensors stems from the following statement: More specifically, the statement: Namely, given $B: V \times W \to U$ and $\xi: U \to \mathbb{R}$, $\xi ...
0
votes
0answers
47 views

Differential forms and minor expansion, question about notation.

There are lectures by Theodore Shifrin on differential forms, and sadly one video ends suddendly where he explains some notation. I try to formulate it in my own words: When k=n, we have ...
0
votes
1answer
57 views

Interpretation of $(r,s)$ tensor

A tensor of type $(r,s)$ on a vector space $V$ is a $C$-valued function $T$ on $V×V×...×V×W×W×...×W$ (there are $r$ $V$'s and $s$ $W$'s in which $W$ is the dual space of $V$) which is linear in each ...
2
votes
1answer
75 views

relative sign in Hodge star of tensor product

Let $V$ be a vector space of arbitrary (finite) dimension and let $(V, \langle \ ,\ \rangle, I) = (W_1, \langle\ ,\ \rangle_1, I_1) \oplus (W_2, \langle\ ,\ \rangle_2, I_2)$ be a direct sum ...
0
votes
0answers
61 views

Zero set of finitely many polynomials.

Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed ...
0
votes
1answer
82 views

Caracterization of isometries that preserve time-orientation in $\Bbb L^3$

First of all, I'm considering $\Bbb L^3$ with the convention: $$\langle (x_1,y_1,z_1),(x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ Let $\Lambda = (\lambda_{ij})$ be an isometry of $\Bbb L^3$. I ...
1
vote
1answer
93 views

R-linear functionals on manifolds

Surely the following is well known: Let $X$ be a (differentiable) manifold, $R$ the ring of continuous/smooth real functions on $X$, $V$ the $R$-module of all continuous/smooth vector fields on ...
0
votes
0answers
29 views

Matrices with functions as entries

I am interested is studying matrices which have functional entries. Specifically I am looking at quadratic forms of the type $x^T Q(x) x$ where $Q(x)$ is a matrix whose entries are functions of $x$. I ...
2
votes
2answers
31 views

Basic geometry proof about tetrahedron

Suppose tetrahedron ABCD such that AB is orthogonal to CD and AC is orgthongal to BD. Show that AD is orthogonal to BC. So i made a picture of a tetrahedron in 3 space and sort of look down at it ...
2
votes
1answer
56 views

Linear dual of vector fields

Suppose that $M$ is a smooth manifold and $\mathfrak{X}(M)$ is the set of smooth vector fields on $M$. There are basically two different linear structures on $\mathfrak{X}(M)$: 1.) $\mathfrak{X}(M)$ ...
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
1answer
70 views

Geometric interpretation of Laplace's formula for determinants

Coming from the geometric point of view, the determinant of an $n \times n$-Matrix computes the volume of an parallelepiped spanned by the columns of the matrix. In context of this question, let the ...
1
vote
1answer
49 views

Topological subspace in $(S^{1})^{n}$

Studying the set of solutions of a particular linear system associated to a matroid, I notice that is it possibile to determine the topology of the quotient and identify it as a subtorus of ...
1
vote
2answers
95 views

Idempotent operators over the exterior algebra

I am wondering if there exists a (reasonably) well-known set of operators $A_i$ over the exterior algebra such that $\{A_i,A_j\} = \frac{1}{2}(A_i +A_j)$, where $\{X,Y\}=(XY+YX)/2$.
6
votes
1answer
74 views

Proof behind $S^n\cong SO(n+1)/SO(n)$

I have been trying to understand the fact that $S^n \cong SO(n+1)/SO(n)$. I believe I have the intuition correct at this point; consider the case when $n=2$ as we have $S^2 \cong SO(3)/SO(2)$.: We ...
1
vote
2answers
57 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
40 views

Difference between flat of a vector and dual of a vector

The flat of a vector $X\in T_p(M)$ is defined as a dual vector $X^\flat\in T_p^*(M)$ given by the following map on vectors: $$ Y\stackrel{X^\flat}{\mapsto} g(X,Y) $$ The dual of a basis vector $e_j$ ...
0
votes
1answer
23 views

Linear transformation from endormorphism to real number

For a finite dimensional vector space $V$, is there a linear transformation between its endomorphism and real number, please? I suspect that since the element of the endomorphism can be represented by ...
1
vote
0answers
33 views

Invariants of shape operators?

Let $S:V\longrightarrow V$ be a Linear Transformation, then the Characteristic polynomial of $S$ and therefore its Coefficient are fixed. Except the first and the last Coefficient that we know their ...
7
votes
3answers
244 views

Isomorphisms between a vector space and its dual

For finite dimensional vector spaces $V$ and $W$, let $i_V: V \rightarrow V^{**}$ and $i_W: W \rightarrow W^{**}$ be natural isomorphisms. Show that for any linear transformation $f : V \rightarrow ...
4
votes
1answer
78 views

Levi-Civita connection compatible with Riemaniann and Pseudo-riemaniann metric

Given a Pseudo-riemaniann metric on ${\cal{M}}$, is it possible to find a Riemaniann metric on ${\cal{M}}$ with the same Levi-Civita connection? If in general this is not possible, what sufficient ...
3
votes
1answer
57 views

tangent/normal space to set of symmetric isospectral matrices

Let $\Lambda = \{\lambda_1, \ldots, \lambda_n\}$ be a set of $n$ distinct real numbers. $M_n(\mathbb{R})$ denotes the set of all $n \times n$ real matrices, and for $B\in M_n(\mathbb{R})$, $B^T$ ...
4
votes
4answers
165 views

Some aspects of inner products in $\mathbb R^3$

I read several descriptions about inner products recently due to an exercise (here it is no longer active so I like to ask again). I am still very confused about this concept. Any clarification will ...
0
votes
0answers
28 views

How to prove a parallel $(u=u_0) $it self curvature?

My name is Gita, and I had aproblem with my math. and need help, I know that a parallel $C$ in a surface of revolution in $M$ be a geodesic if and only if $f'(u_0)=0$. and $C$ is non arc lenght ...
2
votes
1answer
127 views

Surjectivity of an integration map

N.B.: Thanks to studiosus answer I realised I should ask for more conditions or otherwise the answer is straightforwardly wrong. I rechecked my problem and added new assumptions that I boldface. ...
1
vote
2answers
26 views

Proof two solutions of a differential equation are linear independent

Given two solutions for a second order diferential equation: $y(x)=e^{a x}$ and $y(x)=x e^{a x}$ How to show these are linear independent? I procede as follow applying the definition of linear ...
1
vote
2answers
44 views

Dual space (Wikipedia)

I am struggling to understand something on Wikipedia: ''If $V$ consists of the space of geometrical vectors (arrows) in the plane, then the level curves of an element of $V^*$ form a family of ...
0
votes
0answers
19 views

When is a function of two variables positive?

There are two functions, $u(t)$ and $v(t)$, $u:[0,\infty]\to {R}$ , $t=\frac{1}{2}(y_{1}^{2}+y_{2}^{2})$, $v=\frac{-uu'}{2tu'-u}$. They should satisfy the next: $u>0$ and $u+2tv>0$. If we ...
3
votes
1answer
62 views

About a curious cross-product/determinant identity

Whilst proving the fact that one definition of area for a domain inside a parameterisation of some surface embedded in $\mathbb{R}^3$ is well defined, my lecturer made a claim "by linear algebra" that ...
2
votes
4answers
85 views

Boosts in Lorentz-Minkowski space $\mathbb{L}^3$ (or $E_{1}^3$) (+ material)

Can someone give me examples of boosts in $\mathbb{L}^3$? I understand that boosts are isometries that leave pointwise fixed a straight line $\mathcal{L}$. The only thing I can think of, until now, ...
0
votes
0answers
15 views

Constructing a smoothly varying basis without singularities

I am trying to construct a smoothly varying and a differentiable basis to map a vector in $\mathbf{B}:\mathbb{R}^3 \to \mathbb{R}^3$. Given a vector field $\mathbf{n}(\mathbf{x})$ where $\mathbf{n} = ...
3
votes
0answers
147 views

How can higher-dimensional projection maps be described mathematically?

New question: (resulting from discussions with Sabyasachi) I am wonder how can higher-dimensional projection maps, analogous to for example the Mercator, Miller, Behrmann projections, can be ...
21
votes
0answers
697 views

How to find eigenvalues and eigenvectors of this matrix

Can you help to find eigenvalues and eigenvectors of the following matrix? Here is the matrix: $$C = \small \begin{pmatrix} -\sin(\theta_{2} - \theta_{M}) & \sin(\theta_{1} - \theta_{M}) & 0 ...
1
vote
1answer
40 views

the “unit speed” anlogue of the evolute of the curve

Given a curve, $\gamma: \mathbb{R} \to \mathbb{R}^2$ define the flow in the normal direction by $\gamma(t) + \epsilon \, \mathbf{n}(t)$. This is different from the evolute which moves at speed ...
0
votes
1answer
37 views

parameterized ellipse, error in proof of a theorem?

A question from the book "Elementary Differential Geometry" from A Pressley Consider the ellipse $\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$, where $p>q>0$ The eccentricity of the ellipse is ...
1
vote
1answer
26 views

How to show the pushforward is linear using equivalence classes of curves?

Let $M$ be a $C^k$ manifold of dimension $n$. I've constructed the tangent space at $a \in M$ as follows: first I've introduced the following equivalence relation in the set of maps $\gamma : ...
0
votes
2answers
45 views

Distorted Unitary matrices

Let $U$ be an unitary and $D$ be a diagonal matrix. We know that for all vectors $v$ on the sphere $Uv$ is on the sphere and, $$\langle Uv,Uv\rangle=\langle v,v\rangle.$$ What are the vectors $v$ on ...
2
votes
2answers
149 views

Grassmannian, Plucker coordinates

In which books can I find something about the grassmannian and the plucker coordinates ?
0
votes
0answers
64 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
0
votes
1answer
37 views

Characterizing non-degenerate subspaces of Minkowski space

I am trying to show the following equivalence: Proposition 1. A subspace $V$ of the Minkowski space $\mathbb{R}_1^{n+1}$ is non-degenerate if and only if $\langle v, v \rangle \ne 0$ for all $v \in ...
2
votes
0answers
29 views

Existence of maximal tori in infinite-dimensional Lie groups

Reading about Lie groups and maximal tori I came up with a lemma that states that any Lie group $G$ has maximal tori. The proof goes like this: firstly, it is proven that if $H \subset G$ is a proper ...
1
vote
1answer
43 views

Local coordinates for two riemannian metrics

Let $(M,g)$ be a Riemannian manifold, $g' = g + f$ be another metric. Is it possible to get local coordinates such that at a point $P \in M$, $g_{ij} = \delta_{ij}$ and $f_{ij} = 0$ for all $i \not = ...
1
vote
0answers
54 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 ...
2
votes
2answers
72 views

Cochains: terminology

Let a real, smooth manifold $M$ be given. Let $C_k(\mathbb Z, M$) denote the set of $k$-chains with integer coefficients, and let $C_k(\mathbb R, M)$ denote the set of $k$-chains with real ...
2
votes
1answer
124 views

Inner product between tangent vectors

Suppose we are given a point $p$ on a Riemannian manifold $M$ with a tangent space $V=T_{p}M$. Furthur assume that this tangent space is Euclidean. Suppose we have two tangent vectors represented in ...
1
vote
0answers
42 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 ...