Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Smooth chart in what sense?

I have a question concerning smooth manifolds. As far as I've understand a smooth manifold is a pair of a manifold and a smooth atlas. Where smooth atlas means that the transition functions defiened ...
3
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1answer
24 views

Diameter of the Grassmannian

Just an interesting question that came to my mind while studying(!): Since the Grassmannian $G(k,\mathbb{C}^n)$ is a compact manifold, what do we know about its diameter? Do we know any estimate? ...
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0answers
19 views

How to construct an atlas to $T^2$ manifold or torus? [on hold]

The torus embedded in $\mathbb{R}^3$ is $$ T^2=\left \{(R+r\cos(x)\cos(y),R+r\cos(x)\sin(y),r\sin(x))\;| \;x,y \in [0,2\pi),0<r<R \right \} $$
3
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0answers
27 views

Cartan-Maurer trivializations and Bi-invariant Metric on Lie Group

Let $G$ be a Lie group. Let $\nabla^L$ and $\nabla^R$ be the connections on $TG$ corresponding to the trivial connection $d$ on $G\times\mathfrak{g}$ under the left and right trivializations. How ...
0
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1answer
15 views

Length of curves with same images

From a geometrically intuitive point of view, it is obvious that if two injective $C^1$ curves $\gamma,\delta$ with values in $\mathbb R^n$ have the same images, then their lengths $\ell(\gamma)$ and ...
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29 views

Riemannian metric of $3$-sphere

I know this probably seems like a dumb question, I have parametrised part of the unit $3$-sphere with $(x,y,z)\to (x,y,z,(1-(x^2+y^2+z^2))^{\frac{1}{2}})$ and now I'm trying to calculate the ...
0
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1answer
33 views

How do I construct a chart to a infinitely long cylinder embedded in $\mathbb{R}^3$?

An infinitely long cylinder $M$ given by its embedding in $\Bbb R^3, M=\{(R\cos(x),R\sin(x),t)\mid x \in [0,2\pi),t \in(-\infty,\infty),R>0\}$. At least one chart has to be constructed which maps ...
0
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0answers
24 views

Lie algebras and left or right invariant vector fields

Let $G$ be a Lie group and $\phi$ be a diffeomorphism defined by $\phi(\sigma)=\sigma^{-1} $, $\sigma\in G$. I have to prove that $X\mapsto d\phi(X)$ gives an isomorphism of algebras between the ...
2
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1answer
17 views

Typo in theorem about distance square function to a curve

In the following let $\gamma: \mathbb R \to \mathbb R^n$ be a smooth curve and let $u \in \mathbb R^n$. Define the distance square function $f_d: I \to \mathbb R$ on $\gamma$ from $u$ as $f_d(t) = ...
2
votes
1answer
31 views

Prove that $X=\{(x,y,z,t) | x^2+6y^2+4z^2+t^2=1 \}$ is compact and connected

Let be $X=\{(x,y,z,t) | x^2+6y^2+4z^2+t^2=1 \}$. I have proved that $X$ is a submanifold of $\mathbb{R}^4$ of dimension $3$. I have to prove that $X$ is compact and connected. My idea, thinking of ...
3
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0answers
7 views

Is there a connection between topological mixing and squashing functions used in neural networks?

Sigmoid, ReLU, tanh, logistic -type "squashing" functions are popular in neural networks to introduce nonlinearity into the transformations of the input vector, allowing the network to fit complex ...
1
vote
2answers
46 views

Dual tensor for partial derivative, if it has any meaning

I'm trying to find out some details about tensors, so my question maybe isn't quite correct. What if $\omega$ is volume form in $(x,y,z)$ coordinates, then how to understand that ...
1
vote
0answers
50 views

What's the name of this theorem?

If $g: \mathbb R \to \mathbb R^n$ issmooth function and $g^{(i)}(t)=0$ for $1\le i \le k-1$ and $g^{(k)}(t) \neq 0$ then there exists a smooth map $f: \mathbb R \to \mathbb R^n$ such that $g(x) = ...
0
votes
0answers
20 views

What is a “multiple point of contact” of an ellipse and a circle

I'm studying an example which illustrates how to use functions defined on curves to investigate the geometry of a curve: Consider an ellipse $x^2 + 4y^2 = 4$ given in parametrised form $(2 \cos t, ...
1
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0answers
20 views

Finding integral submanifold passing through the origin

I'm having a little trouble with this problem for Lee - Introduction to Smooth Manifolds (2nd ed). The problem is as follows (Problem 19-5): Let $D$ be the distribution of $\mathbb{R}^3$ spanned ...
1
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0answers
17 views

Particular function in proof of flow box theorem

Flow Box Theorem If $M$ is a manifold of dimension $n$ and $X$ is a vector field on $M$ such that for a certain $p\in M$ $X(p)\neq0$, then there exists a chart $(U,\phi)$ on $M$ such that ...
8
votes
2answers
31 views

$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$

How do I write out $\text{Alt}(\phi_1 \otimes \phi_2 \otimes \phi_3)$ for $\phi_1, \phi_2, \phi_3 \in V^*$?
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0answers
11 views

Local conformal coordinates on a surface

Let $\mathcal{M}\subset\mathbb{R}^3$ be a smooth enough regular surface. We want to show that around a point $p\in\mathcal{M}$ there is a neighborhood about $p$ in $\mathcal{M}$ which is parametrized ...
2
votes
1answer
25 views

What is the formal name for the conformal laplacian?

\begin{align} L=R-4\dfrac{n-1}{n-2}\nabla^k\nabla_k \end{align} What is the formal name for $L$? I have seen it referred to as the conformal laplacian, however I thought I once read $L$ with a formal ...
0
votes
1answer
20 views

Confused about tangents to parametrised curves: $y/x \neq y'/x'$

In the following let $\gamma : \mathbb R \to \mathbb R^2$ denote a smooth curve. While trying to derive the equation for the tangent line at the point $\gamma (t)$ I got confused: Observation 1 I ...
4
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0answers
50 views

Understanding composition of vector fields

I've finished a first course on differential geometry and I still find it confusing on how to compose/multiply two vector fields. Let's assume that $X$ and $Y$ are two vector fields on a smooth ...
0
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0answers
30 views

Parallel $ (1,1) $-forms on compact Kähler manifolds.

Let $ (X,\omega) $ be a compact Kähler manifold. We know one example of a parallel $ (1,1) $-form, namely, $ \omega $ itself. Are there obstructions for the existence of non-vanishing parallel $ (1,1) ...
3
votes
1answer
31 views

Gauss-Bonnet Theorem, External Angles and Orientation

The Global Gauss-Bonnet Theorem states: Let $R\subset S$ be a regular region and $C_1,\ldots,C_r$ be closed, simple, piecewise regular curves forming the boundary of $R$. Suposse $C_i$ is positively ...
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votes
1answer
15 views

Preservation of the cross product by parametrization

Let $S$ be a regular surface and $X:U \subset \mathbb{R}^2\longrightarrow X(U)\subset S\subset \mathbb{R}^3$ a local parametrization. Does the following hold? If $e_1, e_2$ are two linearly ...
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1answer
17 views

Proving maps conformal via a scaling factor

I'm in a differential geometry class and I just attended a review session where the TA gave an example problem about conformal maps on the board: Find a constant $k$ such that $x(u,v) = ...
3
votes
1answer
34 views

Gradient vector derived from the metric tensor

According to Frankel's book "The Geometry of Physics", the components of a contravariant gradient vector can be obtained from the inverse of the metric tensor as follows (in section 2.1d, Page 73): ...
0
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0answers
20 views

Surfaces isometric with the sphere [on hold]

Is it possible to find a local isometry between the sphere in $\mathbb{R}^{3}$ and any other surface (not another sphere)? Can you give some examples?
1
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1answer
22 views

Definition of “immersed plane curve”

The exact meaning of immersed plane curve is not clear to me and I would like to request some help with clarifying it here: A plane curve is a map $f: \mathbb R \to \mathbb R^2$. For example, the ...
1
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1answer
28 views

Prove that $\mathbb{R}^2 \times S^1 $ and $M=\left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = 1 \right\rbrace$ are diffeomorphic

Let be $M= \left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = 1 \right\rbrace$. I have proved that $M$ is a embedded submanifold of $\mathbb{R}^4 $ of dimension $3$. I have now to ...
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0answers
14 views

How to find the limit of this flow $\lim_{t \rightarrow \infty} \phi^i_t(p)$ defined by a vector field?

Could anyone help me with how to begin to solve the following problem? We have: Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ ...
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votes
1answer
24 views

How to show that for every element $g$ in a Lie group, the curve $\gamma(t) = g\gamma_v(t)$ is an integral curve such that $\gamma(0) = g$?

Could anyone help me a little with how to begin to solve the following problem? Thanks in advance! Definitions Let $G$ be a Lie group. For $g \in G$, define a diffeomorphism $l_g$ of $G$ by ...
6
votes
2answers
40 views

Properties of the category of smooth vector bundles over a smooth manifold

I am wondering if there are any sources that discuss the properties of the category of vector bundles over a smooth manifold. It seems that most differential geometry texts I've looked at avoid ...
0
votes
1answer
23 views

How to find the determinant of an operator?

I want to learn to find the determinant of an operator. I am given an operator like $\Sigma _{\alpha\beta}=-k^2g_{\alpha\beta}+i\theta\epsilon_{\alpha\beta\gamma} k^\gamma$ Determinant is found using ...
1
vote
2answers
35 views

Expressing a metric as a sum of (possibly) many squares

Given a Riemannian manifold $M$ whose metric $g$ has zero curvature, it is known that we can find local coordinates $x^i$ such that $$g=\sum_{i=1}^{\dim(M)}(dx^i)^2.$$ Conversely, if the curvature ...
1
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1answer
30 views

Is there an algorithm for parametrization of equations?

In this and this Math.SE questions askers wanted to parametrize their equations. It seems to me that one has to, without the algorithm, figure out a symbolic trick and then symbolically manipulate ...
3
votes
1answer
96 views

How could a group be a manifold?

For example a Lie group is defined as a certain differentiable manifold, but what does this mean geometrically, and what is gained by viewing something abstract and algebraic as a manifold? First, I ...
5
votes
1answer
68 views

Can a trefoil knot be stretched to look like a triangle with three knots at the vertices?

Can a trefoil knot be stretched to look like a triangle with three knots at the vertices, like in the right side of the image below, or is that transformation impossible to happen? If possible, what ...
2
votes
1answer
79 views

Is $S^1 \times S^1$ really a torus?

Consider a function $f(x)$ that is $2\pi$ periodic. Consider another function $g(y)$ that is also $2\pi$ periodic. If I wanted to compute the integral of either of these functions I would do so ...
1
vote
1answer
32 views

Retraction to the Boundary on Compact Manifold

I was given the following question on an exam today, "Suppose that $M$ is a compact $n$- dimensional oriented manifold with corners. A retraction to the boundary is a continuously differentiable map ...
-3
votes
1answer
77 views

Can a torus have a simple map from two dimensions to three dimenions like a Gauss map? [on hold]

There are several ways to project two dimensions onto a Riemann Sphere and the Gauss map works very well. A Gauss map: 2d {x,y}-> 3d {2*x/(1 + x^2 + y^2), 2*y/(1 + x^2 + y^2), (1 - x^2 - y^2)/(1 + ...
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2answers
23 views

Properties of the ring of smooth function germs, question on proof.

Let us denote by $C_n$ the ring of $C^{\infty}$ smooth function germs $f : (\mathbb R^n, 0) \to \mathbb R$ or the ring of analytic functions germs $f : (\mathbb C^n, 0) \to \mathbb C$. Denote by ...
3
votes
1answer
91 views

What is the smallest Euclidean space in which one can embed a given curved space?

Given a $d$-dimensional curved space, how many dimensions are required to embed it? As an example think of a sphere's surface, which is a two-dimensional curved space that can be expressed in ...
2
votes
1answer
45 views

Poincaré lemma on a space with trivial homology group

Today I read about Poincaré's lemma from do Carmo's book Differential Forms and Applications. It says that A closed differential $k$-form on a contractible space is exact. I wonder if the ...
5
votes
1answer
28 views

Are the torsion elements dense in every compact Lie group?

Let $ G $ be a compact connected real Lie group. Denote by $ T $ its set of torsion elements. Is $ T $ always dense in $ G $?
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0answers
10 views

Line Elements for $n$-dimensional hyperspheres

I'm currently in the process of deriving the components of the Riemann Curvature Tensor for a 3-sphere using the Cartan Equations. The line element I'm starting with is: $$ ds^2 = ...
0
votes
0answers
30 views

The inverse image of any regular value is a submanifold

Let $f:M^m\subset\mathbb R^k\to N^n$ be a smooth map between manifolds with $m>n$. Let $y\in N$ be a regular value for $f$. Let $x\in f^{-1}(y)$ and $L:\mathbb R^k\to\mathbb R^{m-n}$ be a linear ...
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0answers
17 views

Definition of frontal map

In the lecture about singularities of curves and surfaces the lecturer gave the following defintion: A smooth map $f: U \subseteq \mathbb R^n \to \mathbb R^m$ is called frontal if and only if for ...
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0answers
32 views

Reference to study moduli spaces

I would like to know about references where the problem of finding the infinitesimal deformations of a given geometric structure, and obtaining the corresponding (elliptic?) complex parametrizing the ...
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1answer
56 views

Is $C = \{(x,y) \mid x^3 + xy + y^3 = c \} \subset \mathbb{R}^2$ an embedded submanifold of $\mathbb{R}^2$?

The problem As a continuation of this question (where it was shown that $C$ was a closed $1$-dimensional submanifold for $c \neq 1/27$), I'm trying to find out whether or not $$C = \{(x,y) \mid x^3 + ...
0
votes
3answers
20 views

Greatest area using a string with the length of $l$

Suppose we have a string with length of $l$ what is the shape that has highest area? In other words,with a constant perimeter of $l$ what is the shape with the highest area? P.S:My own speculation ...