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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Lifting of a Diffeomorphism to an Orientable Double Cover

Let $M$ be a non-orientable smooth $d$-dimensional manifold. Let $\tilde{M}$ be the oriented double cover for $M$; and let $f\in Diff^{1+\beta}(M)$. Define $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ ...
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Differential geometry

If we have integrable distribution D of rank k on a manifold, and we have k functions which are constants on the associated integral manifolds. can we glue together these functions to obtain global ...
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30 views

How to compute $[\dot c, X]$ on a manifold?

Consider a smooth curve $c : [0,1] \to M$ and $X \in \mathcal X (M)$. How can one obtain an explicit formula for $[\dot c, X]$? I know the theoretical approach: for every $t \in [0,1]$ there exist a ...
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How do I find a smooth map from complex Gr(k, n) to real Gr(2k, 2n)?

I am trying to find a smooth bijective map from complex Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ to the Grassmannian $2k$-dimensional subspaces of $\mathbb{R}^{2n}$, but I don't ...
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Rotating curve sweeping constant negative Gauss curvature surface

A short line segment rotates around unit circle radius $a$ so that latitude equals longitude or, $ v = u $ so the in the neighborhood of "equator" $ (u\approx a) $ Gauss curvature $ K \approx ...
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Vanishing Integral of a differential form without using Stokes' Theorem

In $\mathbb{R}^3$ consider following 2-form given by $$\omega = xy \: dx \wedge dy + 2x \: dy \wedge dz + 2y \: dx \wedge dz$$ and $$A = \{(x,y,z) \in \mathbb{R}^3 | x^2+y^2+z^2=1, z\geq 0\}.$$ ...
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Curvature Relationship with Norm of the Curve at the Point of Maximum Norm

Continuing on my series of questions, for those following, is the following question: Let $\alpha: I \to \mathbb{R}^3$ be a regular curve. Suppose that for some $t \in I$ the distance from the ...
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Differentiable, Parametrized Curve for Trace $y = |x|$

my professor gave practice problems for an upcoming midterm, and one is to find a differentiable parametrized curve with trace $y = |x|$ with $-1 \leq x \leq 1$. My thinking is that I should find a ...
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The principle curvature of the half cylinder $\{(x,y,z): y^2 + z^2 =1, z>0\}$

I have the half cylinder $\{(x,y,z): y^2 + z^2 =1, z>0\}$ and I want to calculate the principle curvature of this. I know the principle curvatures of this cylinder S at a point p , denoted $k_1$ ...
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Physical meaning of cofactor and adjugate matrix

I like the way there a physical meaning tied to the determinant as being related to the geometric volume. Since the determinant can be calculated through Laplace's formula where the cofactor matrix is ...
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1answer
54 views

$\mathbb{R}^3$ not diffeomorphic to $\mathbb{R}^3\setminus \{0\}$

I have to show, that $\mathbb{R}^3$ is not diffeomorphic to $\mathbb{R}^3\setminus \{0\}$. That means, I have to show that there are no two smooth maps $f\colon\mathbb{R}^3\rightarrow ...
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36 views

Angle between two curves on a surface

Let $\mathcal{M}$ be a surface and $\gamma_1, \gamma_2$ two smooth curves contained in $\mathcal{M}$ in natural parameterization s.t.: $\gamma_1(0)=\gamma_2(0) = p$ , ...
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1answer
48 views

resources for classical gauge theory

As a prospective grad student, I would like to get an entry level introduction to classical (i.e. non-quantum) gauge theory. Please direct me to resources suitable for a novice.
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61 views

Proving a compact Lie group admits a biinvariant metric [duplicate]

At the end of a lesson in Differential Geometry, my teacher said: Fatto, che non dimostriamo, non è difficile ma il tempo scarseggia, se $G$ è compatto possiamo sempre trovare una metrica ...
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1answer
14 views

Is integral curve a embedded 1 dimensional submanifold of the given manifold?

I can easily see a proof that shows its going to be an immersed submanifold . (I am removing the case if the vector field at that point is 0). I am not able to see if it's a embedded submanifold or ...
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singular $1$ cube - Boundary of $2$ chain

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
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How to find the tangent space of a general submanifold?

Given a submanifold $(S,\phi)$ of a manifold $M$, how do we find the subspace of $T_pM$ that is equal to $T_pS$ for $p\in \phi(S)$. I know how to do it for level sets. Is there a way for general ...
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Write $\gamma(t) = (t,t^2,t^3)$ as a graph and a level set

My exercise is to write the twisted cubic as a graph and a level set. However, I am not sure what they mean by a graph and level set. Can anyone explain this please? Do they mean, for a graph, ...
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20 views

Every non-constant closed curve has positive period

I want to show that every non-constant closed curve has positive period, but i'm not really sure how to do this. A smooth curve $r(t): \mathbb{R} \to \mathbb{R}^n$ is $T$-periodic if $r(t+T)= r(t)$ ...
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Independence of parametrization in defining integral of differential form

This is an exercise from Spivak's Calculus on Manifolds. Questions asks the following : (Independence of parametrization). Let $c$ be a singular $k$ cube and $p:[0,1]^k\rightarrow [0,1]^k$ be ...
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Einstein notation

I'm confused about a specific issue that I have with the Einstein notation (for tensor fields on manifolds). I want to write the following thing: Let $X$ be a smooth manifold. Choosing local ...
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1answer
16 views

Conformal curvature line parametrization

While reading a paper I found a definition which is confusing me. Def: A conformal curvature line parametrization $(x,y) \to F(x,y)$ is called isothermic. I know what a conformal ...
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26 views

moving frame with maple

I have already ask this question on stackoverflow, but since it concerns as mathematics than computer science, I ask it here too. I would like to make a classical computation using maple. I would ...
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31 views

Proof of relation between normal of a surface and principle curvatures of surface.

If F(x,y,z) is a scalar function. Then how to prove that, $$\nabla . n = K_1 +K_2$$ where n is normal to surface of constant $F$ given as $$n=\frac{\nabla F}{|\nabla F|}$$ $K_1$ and $K_2$ are ...
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Computing Gauss's of a sphere

The vector field given as $\vec{F}=\frac{\left \langle x,y,z \right \rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}$ The region $D=\left \{ a^{2}\leq x^{2}+y^{2}+z^{2}\leq b^{2} \right \}$ I've some ...
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65 views

When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
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41 views

When is the metric completion of a Riemannian manifold a manifold with boundary?

Let $(M,g)$ be a connected smooth Riemannian manifold and denote by $(M,d)$ the induced metric space following by taking topological metric to be the infimum over length of curves in the standard way. ...
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1answer
29 views

“vector” vs “point” in definition of directional derivative

Given a function $f\colon \mathbb R^n\to\mathbb R$, and given $x,v\in\mathbb R^n$, it is customary to define the "directional derivative of $f$ in the direction $v$ at the point $x$" by $$ D_v f(x) = ...
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46 views

For which Lie groups $G$ can one write $g$ as the exponential $\exp X$ of some $X \in {\frak g}$ for every element $g \in G$?

I am reading a book on matrix Lie algebras (Brian Hall's). Corollary 2.30. says that if $G$ is a connected matrix Lie group, then every element $A$ of $G$ can be written in the form ...
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geometric meaning of conjugate points

Recently I am reading Manfredo do Carmo's Differential Geometry of Curves and Surfaces. He said the $q$ is the conjugate point of $p$ with respect to a geodesic $\gamma$ joining the two points if ...
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Proof that the special linear group $\mathrm{SL}(n,\mathbb{R})$ is a smooth manifold

This is my definition of a smooth manifold I am supposed to work with: Let $\mathcal{M} \subseteq \mathbb{R}^{n}$. The set $\mathcal{M}$ is a $k$-dimensional smooth submanifold of $\mathbb{R}^{n}$ ...
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1answer
45 views

Equivariant generalization of $\mathcal{O}(1)$ on $\mathbb{CP}^1$

The connection of the line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^1$ is given by \begin{equation} A=\frac{i}{2}\frac{\overline{z} \, dz-z\,d\overline{z}}{1+|z|^2} \end{equation} This follows since ...
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Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: ...
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evaluate this region using gauss's theorem (only using the triple integral 'part')

Evaluate $$\iiint _{D}\vec{\nabla} \cdot\vec{F}\,dV$$ with $$\vec{F}=\left \langle x^{2},y,z \right \rangle$$ $$D=\left \{ \left ( x,y,z \right )|x^{2}+y^{2}+1\leq z\leq 5 \right \}$$ ...
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Proving $\mathrm{GL}(n,\mathbb{R})$ is a smooth manifold

Consider the set $\mathrm{GL}(n,\mathbb{R}) = \{ \ A \in M_{n \times n}(\mathbb{R}) \ | \ \mathrm{det}(A) \neq 0 \ \}$. I'm trying to show that this is smooth submanifold of $\mathbb{R}^{n^{2}} \cong ...
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1answer
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Why is even codimension necessary to apply excision for the Euler characteristic?

In this answer on MathOverflow, it is claimed that $$\chi(X/Z)=\chi(X)-\chi(Z)$$ holds for complex subvarieties $Z$ only because $Z$ has even codimension. It is implied that for $Z$ with odd ...
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1answer
33 views

Explicit procedure for integrating densities (twisted $n$-forms)?

I'm aware of several slightly different (yet equivalent) definitions for the density bundle of a non-orientable manifold. Unfortunately, I've been unable to make sense of integration in any of them. ...
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Parametrisation of the curve after a short time

I am trying to wrap my head around this differential geometry problem. I am given velocity V with components in the principle normal and binormal directions. Then I am given an approximation of the ...
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Upper bound on hessian

Given a smooth Riemannian manifold $(\mathcal{M},g)$ and $f \in C^{\infty}(\mathcal{M})$ let $r(x)= d(x,x_0)$ where $d$ is the distance function wrt $g$ and $x_0$ is some point on the manifold. If we ...
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1answer
41 views

Symmetric Curve

I am having hard time solving the following question: Let $\gamma : \mathbb{R} \to \mathbb{R}^2$ curve in natural parameterization. Suppose the curve has curvature $\kappa(s)=3s^2$ Then ...
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1answer
28 views

Darboux coordinates on projective spaces

I am trying to perform some computations in local coordinates on $\Bbb P ^n \Bbb C$ seen as a symplectic manifold, in order to get a better feeling of some facts. While I do know the coefficients of ...
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How to denote raising $x^1$ to a power in differential geometry

I'm working from a text in which the coordinates of a point in $\mathbb R^n$ are denoted $(x^1,\dots,x^n)$. I'm wondering if there is a standard way to denote the sum of the squares of these ...
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How to prove this orthogonality? [duplicate]

In $\mathbb{R} ^3$, $v$ is a fixed vector and $\alpha:I\rightarrow \mathbb{R}$ is a regular curve. We have: $\alpha'(t)$ is orthogonal to $v$ for all $t\in I$ and $\alpha (t_0)$ is orthogonal to $v$. ...
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Does the forgetful functor from smooth manifolds to sets preserve colimits?

It is easily shown that the forgetful functor $F: \mathbf{Man} \to \mathbf{Set}$ preserves limits ($F$ is representable), but does it preserve colimits? It certainly preserves all examples of ...
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Diameter of closed curve is perpendicular to velocity?

Let $\gamma$ be a curve in $\mathbb{R}^2$ and $d:\mathbb{R^2}\times\mathbb{R}^2\to\mathbb{R}$ be the distance function between two points of $\gamma$. I'm trying to show that if the distance between ...
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lower bound on volume of balls

It is well known that a lower bound on Ricci curvature gives an upper bound on the volume of balls. What are conditions that gives a lower bound on the volume of balls? It is reasonable to think that ...
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1answer
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Help with understanding a proof of compact surface having an elliptic point

In my studies of differential geometry from do Carmo's book, I have come across a very nice claim which states that a regular compact surface has an elliptic point that is a point with positive ...
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124 views

Why is $s$ used for arc length?

Why is $s$ used for arc length? I looked around online, but I can't find a definite answer. Thank you!
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What is an integral differential form and how do we recognize it as such?

I am reading about embedding theorems of various types of manifolds (Kodaira's embedding theorem being one of them), and one condition that repeats in all of them is that the manifolds should be ...
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Value preservation along geodesics.

Given a differentiable function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$, where $n>m$. We define the graph of $f$ $ W_f = \{ (x,y) | x \in \mathbb{R}^n, y \in \mathbb{R}^m,y=f(x) \}. $ Given two ...