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 $ surface in 3-space $ $(x,y,z)$ 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) ...
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53 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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Gromov's defenition of Content of Ball

Let $B(p, R)$ denote the metric ball of radius $R$ centered at $p$ in a manifold. Then Gromov defined the content of the ball by $$Cont(B(p,R))=rank(H_*(B(p, R/5))\to H_*(B(p,R))) $$ and he remark ...
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34 views

1-form integration

Let $\alpha:[-1,1]\rightarrow R^2$ be the curve segment given by $\alpha=(t,t^2)$. If $\phi=v^2du+2uvdv$, (the fist component of $R^2$ is $u$ and the second one is $v$) I have $$\int_\alpha ...
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907 views

Shortest path on hyperboloid

On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the ...
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43 views

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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25 views

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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25 views

Is $|K/\tau|\leq 1$?

Let $\alpha:[a,b]\rightarrow \mathbb{R}^3$ be a regular curve such that $\alpha'$ and $\alpha''$ are linearly independent over $[a,b]$. Let $K$ and $\tau$ be the curvature and torsion of $\alpha$ ...
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64 views

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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45 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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30 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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45 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
53 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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29 views

$f(x)=(x^2,x^3)$ 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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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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30 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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51 views

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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23 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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118 views

Instrinsic definition of concave and convex polyhedron

Is it possible to distinguish a concave polyhedron from a convex one by mesurements made only on its surface, without a reference to the 3d space around it?
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25 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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35 views

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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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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Symmetric and wedge product in algebra and differential geometry

I have been struggling with this issue for a while (and asked a similar question here), but still not found a satisfying answer. The question boils down to: which is the correct identity? $dx \, dy ...
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132 views

Torus diffeomorphic to $S^1\times S^1$.

This is an exercise from Guillemin/Pollack's Differential Topology. In a previous exercise, I'm asked to give a complete set of parametrizations of $S^1\times S^1$, which I've succeeded in (I think) ...
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39 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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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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67 views

Geometric characterization of critical points of the Gauss map.

Let $\Sigma \subset \mathbb{R}^3$ an oriented surface by Gauss map $N: \Sigma \rightarrow S^2$. How can I find a geometric characterization of critical points of $N$?
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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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34 views

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

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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96 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 ...
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4answers
301 views

why $\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$ means surface area?

Why the following integral means the area of surface $f(x,y)=z$? $$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$
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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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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
33 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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16 views

Cusp-end in the universal covering

Let $M$ be a n-dimensional hyperbolic manifold with finite volume. Then as a consequence of the Margulis-Lemma we have a decomposition in different types of ends. So let $C$ be a cusp-end. Then there ...
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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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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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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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22 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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1answer
89 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 ...
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16 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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35 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) ...
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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) = ...
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37 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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1answer
23 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 ...