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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“Asymptotic ” transformation for principal curvature lines on positive Gauss curvature K surfaces

If we have lines of curvature for a $ \mathbb R^2 $ $(x,y,z)$ as $ [ f(u,v), \ g(u,v), \ h(u,v) ]$ for negative Gauss curvature surfaces then $ f(u+v,u-v), g(u+v,u-v) , h(u+v,u-v) $ represent ...
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Two surfaces with zero gaussian curvature

There is classical result of Hartman and Nirenberg: Theorem. Every point of a $C^2$ surface of Gauss curvature zero has a neighborhood which admits a parameterization of the form $x=a(u)v+b(u)$ where ...
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Solution of equation with given constraint.

I solved eq. $-\partial^2 A^\nu + M \epsilon^{\nu\rho\sigma}\partial_\rho A_\sigma=0$ Where $A^\nu= \epsilon^\nu(k) e^{ik\cdot x}$ $and$ $ \nu=0,1,2 (2+1 D$ $case)$ and constraints are $k_\mu ...
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Why is the unit normal of plane curves defined to be rotated?

Let $\gamma: \mathbb R \to \mathbb R^2$ be a regular smooth curve given as $\gamma (t) = (x(t), y(t))$ moving at unit speed. Then the unit normal $N$ is defined to be $$ N(t) = (-y'(t), x'(t))$$ ...
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What is the relationship between a complex manifold being Kähler, projective, nonprojective, and nonKähler?

I was wondering if this implication is true. I read a few places that $$\text{nonprojective} \Longrightarrow \text{nonKähler}$$ but I think I maybe have misunderstood. Equivalently, this is of course ...
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Repost (question refined): Maurer-Cartan trivializations and Bi-invariant Metric on Lie Group

Let $H$ be a Lie group. Let $\nabla^L$ and $\nabla^R$ be the connections on $TH$ corresponding to the trivial connection $d$ on $H\times\mathfrak{h}$ under the left and right trivializations: ...
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1answer
19 views

Need reference on indicies of critical points

I came across the term "index-1 critical point" in my reading, and I would like to know if there are some good references to learn about indices of critical points of smooth functions. The wikipedia ...
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1answer
23 views

Derivative of exponential maps in Lie group $G$ and the adjoint operator on its Lie algebra

Let $G$ be a (not necessarily compact, probably even infinite dimensional) Lie group, and $g$ be its Lie algebra. Let $V,W\in g$. Consider $J(t):=(Dexp)_{tV}(tW)$ be the result of differential of the ...
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1answer
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$f(x)=(x^2,x^3)$ not an immersion but $Df$ one-to-one?

Let $f:\mathbb R\to\mathbb R^2$ with $f(x)=(x^2,x^3)$. Then $f$ is not an immersion since $rank Df\neq1$ for $x=0$. Our lecturer told us that this is equivalent that $Df$ is one-to-one. What is meant ...
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Defining functions for connected sets

Let $\Omega \subset \mathbb{R}^n$ an open, bounded and connected set with a $C^2$ boundary and a function $\rho \in C^2(\mathbb{R}^n)$ such that $$ \Omega = \{ x \in \mathbb{R}^n : \rho(x) < 0 ...
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Injectivity of a map on a non convex domain

Let $\Omega \subset \mathbb{R}^n$ open, bounded, and connected, a map $f \in C^1(\Omega)$ and $\alpha > 0$ such that $$ \langle \nabla f(x)\xi ; \xi \rangle \geq \alpha |\xi|^2,\quad \forall\, x ...
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1answer
38 views

Index notation.

I have a very basic question concerning index notation, normally used in physic papers. I have never used index notation and it is very difficult to me to translate from index free notation, even in ...
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1answer
37 views

Points of $4$-contact of an ellipse and a circle

Consider an ellipse $x^2 + 4y^2 = 4$ given in parametrised form $(2 \cos t, \sin t)$. At a given point $p_0 = (2 \cos t_0, \sin t_0)$ we want to measure how round the ellipse is (i.e. how similar to a ...
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Mean Curvature Flow - First Variation Formula

I'm just going through Huisken's paper, flow by mean curvature of convex surfaces into spheres, and I'm stuck on one of the Lemma's proven in the paper. The idea is to prove (for the normalised ...
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1answer
27 views

Cocycle conditions of fibre bundles

In a classical approach of fibre bundle one always need the cocycle condition is satisfied, namely: $$g_{12} g_{23} g_{31}\equiv 1$$ in $U_1\cap U_2\cap U_3$. However, I do not see why this cocycle ...
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Why can't differentiability be generalized as nicely as continuity?

I was a little bit dissapointed when I learned to differentiate on manifolds. Here's how it went. A younger me was studying metric spaces as a first unit in a topology course, when a shiny new ...
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Matrix Calculus and Matrix Derivatives

Consider a map $f : \mathbb R^{n\times m} \to \mathbb R^{p \times l}$ between matrix spaces, what is the differential of such a mapping? I looked at a really simple example, $\operatorname{id} : ...
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22 views

How does the Schrodinger's potential transformer if the metric conformally transformers?

Given Schrodinger's equation $$ (-\nabla^2+V)\psi=E\psi $$ and the conformal transformation $\tilde{g}_{mn}=e^{2\phi}g_{mn}$, how does the Schrodinger's potential $V$ transformer if the metric ...
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28 views

Write the vector field such that its integral curves are the meridians of $S^2$

I have problems in writing down a vector field $X$ on the sphere $S^2$ such that the integral curves of $X$ are the meridians of $S^2$. I proceed in this way. I consider this parametrisation of ...
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Law of Sines from Gauss Bonnet Thm

Can the Law of Sines in Spherical Trigonometry be derived from Gauss Bonnet Theorem? EDIT1: To express what went through my mind so far: Started with 3 geodesic circle arcs ... angles at corners can ...
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26 views

Reversal metric [on hold]

The operation of changing the semi-Riemannian manifold $M$ with metric tensor $g$ to the same smooth manifold with metric tensor $-g$ is called reversing the metric of $M$. I don't understand it. ...
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1answer
33 views

Curvature and convexity of a plane curve

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2$ be a $C^2([a,b])$ regular curve. Is it true that $\mathbf{r}$ is convex if and only if its curvature $\kappa(t)\leq 0, \forall t\in [a,b]$ or $\kappa(t)\geq 0, ...
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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 ...
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1answer
33 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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1answer
17 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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1answer
39 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 ...
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1answer
38 views

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

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 ...
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1answer
20 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) = ...
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32 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 ...
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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 ...
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2answers
59 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 ...
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51 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) = ...
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1answer
22 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, ...
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22 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 ...
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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 ...
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$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$

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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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 ...
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1answer
28 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 ...
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1answer
21 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 ...
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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 ...
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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) ...
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1answer
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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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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
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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
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1answer
35 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): ...
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21 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?
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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 ...
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1answer
30 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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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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1answer
25 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 ...