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
33 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
38 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
67 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
31 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
70 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
37 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
59 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
59 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 $\phi^* f = f \circ \phi : M \to \mathbb{R}$. ...
4
votes
3answers
134 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
57 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
47 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
99 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
66 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
61 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
112 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
74 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
52 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
88 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
203 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
299 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
191 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
215 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
84 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
122 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
214 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 $3\times 3$ ...
0
votes
1answer
37 views

Path contained in Surfaces

$y(t)$ is a path contained in two surfaces: $x^2+y^4+z^6=3$, $x+y^2=y+z^2$ also $y(0)=(1,1,1)$ and $||y'(0)||=1$ Need to find the vectors $-y'(0)$ and $+y'(0)$ To be honest, I'm not sure how to ...

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