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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Area of a submanifold defined by an equation

My purpose is to find the surface area of a torus defined by the equation : $$(\sqrt{x^2+y^2}-a)^2+z^2=r^2$$ where $a,r$ are parameters such that the torus is non degenerated. First I know a mean of ...
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54 views

what does it mean to have inner product of $S^2$ and $R^3$?

It may be that the title of my question is wrong but i am writing this question because i am struck while reading this paper Brownian motion on rotational group Where $^*\mathscr{f} $ is transpose of ...
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13 views

Is striction curve of a ruled surface orthogonal to generators?

Let $M$ be a ruled surface of $R^3$ with a regular parametrization given by: $$x(u,v)=\alpha(u)+v\beta(u)$$ where $α′≠0$ and $\|\beta\|=1$. If I know that $\alpha$ is the striction curve of a ...
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42 views

Zero gradient in $L^2(M)$

I'd like to show that for $u \in L^2(M)$, for M a compact, connected Riemannian manifold, if $\nabla_g u = 0$ (i.e $\forall X$ $C^{\infty}$- vector field on $M$, $\int_M u \hspace{1mm} \text{div}_g X ...
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Does $(\partial_ x)^2=\partial_{xx}$

I am trying to prove the identity $$4 \partial \bar \partial=\partial_{xx}+\partial_{yy}$$ Where $$\partial=\frac{1}{2}(\partial_x-i \partial_y)$$ and $$\bar \partial=\frac{1}{2}(\partial_x+i ...
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33 views

Converse to pullback pasting and local diffeomorphisms

The nlab page on local diffeomorphisms gives the following two equivalent conditions for a smoonth function to be a local diffeomorphism. The derivative is an isomorphism of tangent spaces ...
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20 views

3 circles on a sphere what is the radius of the sphere

On a sphere with radius R there are 3 circles: Circle $C_1$ with radius $r_1$ and circumference $c_1$ Circle $C_2$ with radius $r_2$ and circumference $c_2$ Circle $C_3$ with radius $r_3$ and ...
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2answers
40 views

Embeddings of surfaces into a 3-manifold

Say we are given a disconnected closed orientable surface $S=S_1\coprod S_2$ with $f=f_1\coprod f_2:S\rightarrow M$ such that the $f_i$ are embeddings and the images are incompressible. Suppose that ...
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1answer
18 views

Effect of mapping of principal fiber bundles on a principal connection

Let $\lambda = (P,\pi,M,G)$ and $\lambda' = (P',\pi',M',G')$ be two principal fiber bundles, $\lambda$ being equiped with a principal connexion whose connexion form is $\omega$. If $(f,k,\rho)$ is a ...
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21 views

Book or lecture note on presymplectic-manifolds

I'm looking for a book/ lecture note in which presymplectic manifolds and matters relating to them (specially dynamics of Hamiltonian systems) has been fully explained. Does anyone know such a book? ...
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1answer
34 views

A corollary of Li-Yau-Hamilton estimate

Picture below is from the Hamilton's The Harnack estimate for the Ricci flow .How to get the corollary 1.2 by Theorem 1.1 ? It seemly be not immediately and hard to compute. Maybe just because I am ...
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29 views

Find the curvature tensor and sectional curvature associated with the first fundamental form$I=du^2+f^2(u)dv^2$

Consider the surface of revolution $\sigma$ in he Euclidean space $\mathbb{R^3}$ given by $$\sigma(u,v)=(f(u)cosv,f(u)sinv, g(u))$$ with $f>0$ where the profile curve has unit speed. The first ...
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1answer
40 views

Can we lower bound the volume of the image of a ball under a diffeomorphism?

Apologies if this question is overly simple, I'm new to differential geometry. Suppose I have two Riemannian manifolds $M_1$ and $M_2$, along with a diffeomorphism $f:M_1\to M_2$ between them. Let ...
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2answers
35 views

Index attached to derivative operator

So in "General Relativity", Wald introduces a derivative operator $\nabla$ on a smooth manifold $M$ that sends $(k, l)$ tensors to $(k, l + 1)$ tensors. One of their properties he says is that if $f ...
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35 views

**Prove that if all the principal normals of a curve pass through a fixed point, the curve is either a circular arc or a whole circle.**

Prove that if all the principal normals of a curve pass through a fixed point, the curve is either a circular arc or a whole circle. I determined that I would need to prove that torsion was zero and ...
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1answer
30 views

How can I express a waveform or parabola with the peak(s) cut off/flattened?

I want the peaks of the wave or parabola (above a certain positive or negative threshold) to be flattened - to look like mountains or valleys with flattened tops and/or bottoms. Is there a simple ...
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20 views

Existence of solution of integral curve on manifold

Let $M$ be an $n$-dimensional manifold and $X$ be a smooth vector field on $M$. In all books I found that the proof all uses the existence of solution of ODE in $\mathbb R^n$. I try to give an ...
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1answer
27 views

Null-homotopic map from $SL_2(\mathbb{R})$

I need to prove that a smooth map $f\colon SL_2(\mathbb{R})\rightarrow S^4$ is homotopic to the constant map. I think that computing the corresponding homotopy groups may help, but I don't see how to ...
3
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1answer
42 views

Product of inexact differential forms is inexact

Suppose we have a product manifold $M = M_1 \times M_2$. Let $\omega$ be a closed but inexact form on $M_1$ and $\eta$ a closed but inexact form on $M_2$. Then the claim is that $$\omega \wedge \eta$$ ...
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1answer
29 views

arithmetic average over the spherical surface?

intuition behind taking arithmetic average over the spherical surface? . wiki definition :- Consider an open set $U$ in the Euclidean space $R^n$ and a continuous function $u$ defined on $U$ with ...
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1answer
33 views

Compact Poincaré dual of $S^{n-1}$ in $\mathbb{R}^n \backslash \{0\}$

I have been asked to find a sub manifold $S \subset M := \mathbb{R}^n \backslash \{0\} $ s.t. its compact Poincaré dual is a basis of the first cohomology group $H^1_c(M) = \mathbb{R}$. Now, $S$ must ...
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1answer
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Is it true $f(x,y)$ and $af(x,y)$ has the maximum curvature at same point?

Let assume function $f(x,y)$ has the max principle curvature at point $(x_0,y_0)$. And is it true $a\cdot f(x,y)$ ,($a\in\mathbb{R}$) has the max principle curvature at the same point? I think test ...
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1answer
40 views

Components of the metric tensor and its inverse

Let $g$ denote a metric tensor. Then Wald writes (in his book on general relativity): "The inverse of $g_{ab}$...is a tensor of type $(2,0)$ and could be denoted as $(g^{-1})^{ab}$. It is convenient, ...
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42 views

What is the Riemannian metric induced on a surface $M \subset \Bbb R^3$ by the usual flat metric?

Let $D$ be an open subset of $\mathbb{R}^3$, and $f: D \to \mathbb{R}$ be a smooth function whose gradient $ \nabla f \neq 0$ on $D$. Consider the surface $M = \{(x_1,x_2,x_3) \in D \mid ...
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25 views

There is no way to integrate real-valued functions in a coordinate independent way on a manifold

In Lee's ISM, an example is given to show that there is no way to integrate real-valued functions in a coordinate independent way. Suppose $C\subset \mathbb R^n$ is a closed ball, and $f:C\to ...
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25 views

Can the rank of harmonic map decrease far from the boundary?

This question is in some sense a continuation of this question, though it asks something weaker. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are ...
3
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2answers
93 views

How to understand the notion of a differential of a function

In elementary calculus (and often in courses beyond) we are taught that a differential of a function, $df$ quantifies an infinitesimal change in that function. However, the notion of an infinitesimal ...
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Question about definition of coordinate charts

I am having trouble understanding the following question, which I will paraphrase here: The $n$ sphere $S^n$ is defined: $S^n= \left\{ \textbf{x} \in \mathbb{R}^{n+1}: |\textbf{x}|=1 \right\}$. It ...
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1answer
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Finding all $2$-forms in the right half-plane that are invariant under glide transformations

I'm trying to find all 2-forms $\omega$ that are invariant under glide transformations in the right half-plane, $X = \{ (x,y) \in \mathbb{R}^2 : x > 0\}$. To do this, we can write the vector field ...
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1answer
16 views

Show that every curvature of a Frenet curve satisfy the following statement.

I need to show the following statement: Show that for every Frenet curve $c:I\to\mathbb{R}^n$, the curvatures $\kappa_1(t),\ldots,\kappa_{n-1}(t)$ satisfy the following equality: ...
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10 views

Conditions for the trace of a curve to belong to a line

I want to find the conditions so that the trace of the curve $\vec{x}\in\mathbb{R}^3$ belongs to a line. I know that the equation of a plane $\alpha$ in $\mathbb{R}^3$ is given by $ax+by+cz+d=0 ...
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How I could define a inner product in the characters in $SL(2, \mathbb R)$

I have homework in a course on Lie groups, in which I must show that $(\pi_m,\mathbb C_m[x,y])$ are the only irreducible representations of finite dimension in SL(2,ℝ). Here $ C_m[x,y])$ is the vector ...
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24 views

Parametric surfaces in Riemannian manifolds

Let be $(M^n,g)$ a Riemannian manifold and $c: [0,l]\to M$ a geodesic with unit speed. Consider the parametric surface $f$ is $M$, given by $$f(s,t)=\exp_{c(s)}(tn(s)),$$ where $(s,t)\in [0,l]\times ...
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Is this proof about the Lie brackets and flows correct as given?

In this post, Fredrik Meyer gives a proof to the following formula(Please see the conditions and the meaning of notations in the link): $\frac{d}{dt}|_{t=0} \alpha(t) = [X,Y](p)$, where ...
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1answer
25 views

Question on the definition of connection in Liviu.Nicolaescu's book

In "Lectures on the Geometry of Manifolds",Liviu said that a connection on a principle $G$-bundle defined by an open cover $(U_{\alpha})$ and gluing cocycle $g_{\alpha\beta}:U_{\alpha\beta} \to G$ is ...
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2answers
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Proving that $N$ is a manifold.

I'm dealing with the following exercise from Munkres' "Analysis on Manifolds": Let $f:\mathbb R^{n+k}\rightarrow \mathbb R^n$ be of class $C^r$. Let $M$ be the set of all $x$ such that $f(x)=0$. ...
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1answer
32 views

$A\cup \text{Fr}A$ is a manifold

I'm dealing with the following exercise from Spivak's "Calculus on Manifolds": Let $A\subset \mathbb R^n$ be an open set such that the (topological) boundary of $A$, $\text{Fr}A$ is an $n-1$ ...
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25 views

Finding a generating curve for a regular valued surface

Given a regular surface $X=\{(x,y,z):x^2+y^4+z^3=1\}$ and a point $p=(1,1,-1)\in X$ and a tangent vector $u = (2,-1,0)\in T_pX$ define a generating curve $\alpha (t):(-i,i)\rightarrow X$ for $u$ such ...
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28 views

inner product of characters in $SL(2,\mathbb{R})$ [duplicate]

tengo una tarea en un curso de grupos de Lie, en la cual debo demostrar que $(\pi_m,\mathbb{C}_m[x,y])$ son las únicas representaciones irreducibles de dimension finita en $SL(2,\mathbb{R})$. Donde ...
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1answer
35 views

Projectivised tangent bundle of 2 sphere

I'm trying to understand how rotations act on the "projectivised" tangent bundle of the sphere. Let $S^2$ be the two sphere and denote by $P(TS^2)$ the tangent bundle where each tangent space ...
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The curvature is equal to the derivative of the angle between the curve and the x-axis?

I'm trying to prove that if $\vec{x}:I\rightarrow\mathbb{R}^2$ is a curve parametrized by arc length and $\theta(t)$ is the angle between the tangent line to $\vec{x}$ at point $t$ and the $x$ axis, ...
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22 views

Definition of the derivative of a 2nd-order tensor with respect to a scalar

The derivative of the (positive definite, symmetric, 2nd-order) tensor $\mathbf{C}(t)$ with respect to the scalar $t$ is defined as: $$ \frac{\partial \mathbf{C} }{\partial t} = \lim_{\Delta ...
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1answer
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Orientability of $ x(u,v)= \bigg(\bigg(1+v\cos\frac{u}{2}\bigg)\cos(u), \bigg(1+v\cos\frac{u}{2}\bigg)\sin u, v\sin\frac{u}{2}\bigg) $

Consider the map: \begin{equation} x: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}: (u,v) \rightarrow \bigg(\bigg(1+v\cos\frac{u}{2}\bigg)\cos(u), \bigg(1+v\cos\frac{u}{2}\bigg)\sin u, ...
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1answer
67 views

Conditions on characteristic polynomial to define a matrix submanifold.

I'm trying to find conditions on the characteristic polynomial, $p$, of a matrix such that the pre-image of matrices with characteristic polynomial $p$ form a manifold. More precisely, we can write ...
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18 views

Application of Gauss-Bonnet theorem

What's the total curvature of this surface ? I know that I have to find a diffeomorphism between this surface and a sphere or a torus, but I can't find it! I will appreciate your help!
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definition of covariant derivative (along curve)

An affine connection on a smooth manifold $M$ is a map $\nabla: \mathcal{V}(M) \times \mathcal{V}(M) \to \mathcal{V}(M)$ satisfying several properties, where $\mathcal{V}(M)$ denotes the set of smooth ...
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1answer
31 views

Every open cover of a smooth Manifold has a regular refinement

I am trying to understand the proof of Let M be a smooth manifold. Every open cover of M has a regular refinement. The proof begins as follows [Lee] : Let $X$ ...
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1answer
41 views

How to calculate this derivative in differential geometry

Given a symmetric matrix $A$ and a function from generalized linear group to generalized linear group $$f: \text{GL}(n,\mathbb{R})\rightarrow \text{GL}(n,\mathbb{R}), g\mapsto g^TAg$$ For $\forall ...
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1answer
16 views

Is there exist a smooth function $f:\mathbb{R}^{m}\to M$ such that $f|_{U′}=\phi^{-1}$?

Let $M$ topological smooth manifold and $(U,\phi)$ chart fixed with $\phi(U)=U′$ open in $\mathbb{R}^{m}$. Is there exist a smooth function $f:\mathbb{R}^{m}\to M$ such that $f|_{U′}=\phi^{-1}$? I ...
1
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0answers
32 views

Definition of “differential” [duplicate]

I am confused about the definition of "differential". Sometimes I see it is a pushforward mapping $df:TM\rightarrow TN$, which gives another tangent vector when acting on a tangent vector. ...