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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114 views

When is There a Solution to “Pullback Equation” of Differential Forms

All: Let $f: M \to N$ be a smooth map between manifolds, and let $w$ be a $1$-form on $M$. Under what conditions is there a $1$-form $z$ defined on $N$ so that $w=f^*z$, i.e., so that $w$ is the ...
1
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0answers
44 views

Confusion about orientations in Greens second identity

This question has been the source of some confusion on my part so I am hoping there is someone out there who can clear it up. Let $\Omega \in \mathbb{C}$ and $f,g\in C^{\infty}_c(\mathbb{C})$. It is ...
1
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1answer
107 views

Equivalence class of similar triangles

I am trying to understand this particular example in a book I am going through. Consider the set of all equivalence classes of similar triangles , T. The book says that if I apply a similarity to ...
2
votes
2answers
533 views

Pullback of a $1$-form

All: I looked at the list of similar questions, but none seemed to be done explicitly-enough to be helpful; sorry for the repeat, but maybe seeing more examples will be helpful to many. So, I have ...
0
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1answer
48 views

Show that the tangent of $C$ is a normal to the original curve.

Let $r(t)$ be a parametrization of a curve in $\mathbb R^2$ with curvature $K(t)\neq 0$ and normal vector $N(t)$. The parametrization $r_C(t)=r(t)+N(t)/K(t)$ defines another curve $C$. Show that ...
12
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3answers
859 views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
1
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1answer
58 views

The tangent space of $\mathrm{Aut}(T_eG)$

Let $G$ be a Lie group and $e \in G$ be the identity. I want to understand the following sentence. " $\mathrm{Aut}(T_eG)$ being just an open subset of the vector space of endomorphisms of $T_eG$, its ...
9
votes
5answers
340 views

Does this interesting property characterize a sphere?

Consider 2-d surfaces in 3-d (at the suggestion of a comment, let's say closed connected 2-dim smooth manifolds, embedded in dimension 3) with finite area. A sphere has the interesting property that ...
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0answers
30 views

Rotationally symmetric hypersurfaces with mean curvature bounded away from 0

I know that the rotationally symmetric hypersurfaces in $\mathbb{R}^n$ with constant mean curvature are the hyperplane, sphere, cylinder, catenoid, nodoid, and unduloid. Are there any significant ...
0
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1answer
134 views

Exterior Product $d\Phi_1\wedge d\Phi_2$ and spherical coordinates

One short question: If $\Phi\colon\mathbb{R}^3\to\mathbb{R}^3$, defined by $$ \begin{pmatrix}r\\\vartheta\\\phi\end{pmatrix}\mapsto\begin{pmatrix}r\sin \vartheta\cos \phi\\r\sin \vartheta ...
0
votes
0answers
66 views

Points positioned on a surface with maximum distance

Given a spherical shell with area A. I want to arrange n points on this surface in such a way, that the distance between those n points is maximal. Do you know how to do this?(Can we say something ...
11
votes
1answer
210 views

Are there exotic symplectic structures on $ S^2 $?

Besides the obvoius symplectic structure on $ S^2$ given by the area element in the standard embedding $ S^2 \to \Bbb R^3$, are there any other closed 2-forms on $ S^2$ which produce nonisomorphic ...
3
votes
3answers
134 views

Inner product of De Rham cohomology classes

Is there a well-defined inner product between cohomology classes? In particular, is it possible to extend the Hodge inner product? If I try, I obtain this: $$\int *(\omega + d\lambda)\wedge (\sigma + ...
3
votes
1answer
380 views

Show that the arc length of a curve is invariant under rigid transformation.

Show that the arc length of a curve is invariant under rigid transformation. The curve here is in $\mathbb R^3$, and the definition of arc length is $\int^b_a||\bf r'$$(t)||dt$. This theorem ...
4
votes
3answers
369 views

Isotropic Manifolds

Recall that a Riemannian manifold $(M,g)$ is isotropic if for any $p\in M$ and any unit vectors $v,w\in T_pM$ there is an isometry $f:M\to M$ such that $f_\ast(v)=w.$ Recall also that $(M,g)$ is ...
2
votes
1answer
116 views

why positive scalar curvature manifolds

I am studying scalar curvature and I have seen that many mathematicians studied obstruction against positive scalar curvature (for example Stolz, Schick, Roe, J. Rosenberg, Hanke and many others). ...
3
votes
1answer
107 views

Prove $G = \left\{ \mathrm{diag} (e^{ti}, e^{\lambda ti}) \mid t \in \mathbb{R} \right\}$ is not a manifold.

Let $\lambda$ be an irrational number. Let $G \subset G_2(\mathbb{C})$ be defined as $G = \left\{ \mathrm{diag} (e^{ti}, e^{\lambda ti}) \mid t \in \mathbb{R} \right\}$. Prove that $G$ is not a ...
4
votes
3answers
794 views

When does a null integral implies that a form is exact?

It is trivial to prove that the integration of a $(n-1)$ exact form on the boundary of a $n$-manifold is 0. What about the contraposative ? If the integration of a $(n-1)$-form on the boundary of a ...
2
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1answer
85 views

scalar curvature

I am studying scalar curvature. It is the trace of the Ricci operator. I read that its geometric meaning follows from this formula ...
4
votes
2answers
108 views

Why is $\int_{\partial D}x\,dy$ invalid for calculating area of $D$?

I am just learning about differential forms, and I had a question about employing Green's theorem to calculate area. Generalized Stokes' theorem says that $\int_{\partial D}\omega=\int_D d\omega$. ...
2
votes
1answer
180 views

Approximate parallel transport using Jacobi fields

Let $M$ be a riemannian manifold (let $\left\langle \cdot,\cdot \right\rangle_{p}$ be the scalar product on $T_{p}M$). Let $p \in M$ and $\xi \in T_{p}M$. We consider the geodesic $\gamma \, : \, t \, ...
0
votes
1answer
228 views

3D Road - Rotate around 3d curve

First of all, I'm not sure whether to post this on stackoverflow or here, but since there's some mathematics needed here (especially at the end of this question) I posted it here. I'm given a ...
4
votes
2answers
176 views

Distance metric for (infinite) lines in 3D

I would like a metric $d(\;)$ between pairs of (infinite) lines in $\mathbb{R}^3$, with these properties: If two lines $L_1$ and $L_2$ are parallel and separated by distance $x$, then $d(L_1,L_2) = ...
1
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1answer
96 views

Jacobian of matrix product in SU(n)

I need to compute the determinant of the jacobian matrix of the function $f: SO(n)\times SO(n) \rightarrow SO(n)\times SO(n)$ given by $f(P,V) = (PV, P^TV)$. I've found the jacobian if we extend $f$ ...
0
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1answer
69 views

Isometric embeddings with prescribed second fundamental form

I'm looking for some non-rigidity result for isometric embeddings in euclidean space (codimension 2). For example, any isometric embedding of the round $S^2$ into $\mathbb R^3$ is unique up to rigid ...
1
vote
1answer
61 views

One Point Derivations on locally Lipschitz functions

Let $A$ be the algebra of $\mathbb{R}\to\mathbb{R}$ locally Lipschitz functions. What is the vector space of derivations at $0$? The proof that for continuous functions there aren't really any doesn't ...
4
votes
1answer
115 views

Diffeomorphisms preserving harmonic functions

I'm looking for smooth maps $ \Bbb R^n \to \Bbb R^n $ with the property that, whenever $ h $ is a harmonic function ($\Delta h=0$), $ h\circ f $ is also harmonic. Is there a nice characterization of ...
2
votes
0answers
677 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
1
vote
1answer
170 views

Show that along the cycloid, the tangent vector is not well-definded when $\theta=2k\pi$.

Show that along the cycloid, the tangent vector is not well-defined when $\theta=2k\pi$. p.s. : A cycloid has the parametrization $\bf{r}$$(\theta)=(\theta-\sin\theta,1-\cos \theta)$. ...
9
votes
1answer
353 views

Electrodynamics in general spacetime

Let $M\cong\mathbb{R}^4_1$ be the usual Minkowski spacetime. Then we can formulate electrodynamics in a Lorentz invariant way by giving the EM-field $2$-form $\mathcal{F}\in\Omega^2(M)$ and ...
2
votes
1answer
63 views

Quantitative Transversality

I came across the following problem while reading some literature in Dynamical Systems. Say I have an ambient Riemannian manifold $(M,g)$ and a pair of transverse embedded disks $D_1, D_2$ of ...
5
votes
1answer
125 views

Showing every knot has a regular projection using diff top

My question is: Can we use differential topology to prove that every smooth knot has a regular projection? Here is some background: Let $\gamma : S^1 \rightarrow \mathbb{R}^3$ be a smooth unit-speed ...
2
votes
0answers
119 views

Torsion of fundamental group is abelian

On Riemannian manifold with Ricci curvature bounded below (For example, flat torus. It has ${\bf Z}^2$ as fundamental group, which is nilpotent (=almost abelian). In fact Ricci curvature bouned ...
2
votes
0answers
87 views

Momentum map and equivariance

I am reading an article in which I do not understand some equivariance property about the momentum map. Let $G$ be a Lie group acting on a manifold $Q$. The action is denoted $(g,q) \, \mapsto \, q ...
3
votes
1answer
96 views

geodesic metric

I'm trying to prove that the line segment is the minimizer of the distance $$d(x,y)=\inf l(\gamma),$$ where $x,y\in X$, $X$ is a Banach space, $\gamma$ is a path from $x$ to $y$ and ...
2
votes
2answers
96 views

Representing a Riemannian metric in $\mathbb R^3$ restricted to the upper half of $S^2$

Consider $M := \mathbb R^3$ as a smooth manifold with a Riemannian metric $g := \sum_{i=1}^3 dx^i\otimes dx^i$, where $(x^1, x^2, x^3)$ is the standard coordinate of $M$. Let $N\subset M$ be a ...
2
votes
1answer
64 views

Minimizing Surface Curvature

I have a tensor-product $B$-spline surface. I have been able to determine all the control points of the surface so that all the points are given as a function of only one of these points, that we call ...
0
votes
1answer
69 views

When is the restriction of a Lorentzian metric to a regular submanifold degenerate everywhere?

Let $M$ be a $C^\infty$ manifold, $N\subset M$ be a regular submanifold and $g$ be a Lorentzian metric on $M$. I would like to find $M$, $N$, $g$ such that the restriction of $g$ to the tangent ...
3
votes
2answers
99 views

How is the norm on $C^k(M)$ defined?

Let $M$ be a smooth, compact $n$-dimensional manifold and $C^k(M)$ the space of real valued $C^k$-maps on $M$. I am looking for a definition of the norm $|\cdot|_k$ on $C^k(M)$ that induces the ...
3
votes
2answers
120 views

Smooth partitions of unity

Let $ M $ be a Riemannian manifold and let $ \{U_i\} $ be a countable covering of $ M $. It is well known that there exists a countable collection of smooth function with compact support $ \{\rho_i\} ...
2
votes
0answers
107 views

Asymptotics of Green's function of laplacian

Let $ \Omega $ be a domain in $ \mathbb{R}^2 $ with a Riemannian metric $ g $, and let $\Delta $ be the laplace-beltrami operator induced by the metric, with dirichlet boundary conditions. For $ x,y ...
3
votes
2answers
54 views

Problem with simple laplacian equation

I would like to solve the following PDE: $$ \partial_x^2 u + \partial_y^2 u = -\frac{2 x^2 (x^2-y^2)}{\left(x^2+y^2\right)^2} $$ The right side comes from $ x^2 \partial_x^2 \log(x^2 +y^2) $. ...
2
votes
1answer
536 views

Surface area element of an ellipsoid

I would like to evaluate an integral numerically over the surface of an ellipsoid. Take an $N \times N$ grid over the parameter space $(u, v) \in [0, 2\pi) \times [0, \pi) $. A simple approximation of ...
5
votes
0answers
108 views

A good reference for learning about super-differentiation & super-integration?

I've looked at a couple of books for basic information for super-differentiation & super-integration - Rogers Supermanifolds, and Khrennikovs Superanalysis. Unfortunately both books lack a clear ...
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vote
0answers
80 views

Notation in riemannian geometry

I am reading a lecture on Riemannian geometry in which it is written that, for a differentiable manifold $M$ and a differentiable curve $v \, : \, I \, \longrightarrow \, M$ defined on an interval ...
2
votes
1answer
181 views

Prove that a compact cone is not diffeomorphic to the 2-sphere

In Tapp's "Matrix Groups for Undergraduates" he briefly states (p.103) that a compact cone (he just shows a picture of a manifold with a ''cone point'') is not diffeomorphic to a 2-sphere. I would ...
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vote
0answers
146 views

Use Möbius Transformation Normal Form to prove Lambda

I'm just completely lost on how to answer this question: Let $$\frac{Tz-p}{Tz-q}=\lambda \frac{z-p}{z-q}$$ be the normal form of a Möbius transformation with two fixed points. Prove that $\lambda$ = ...
3
votes
2answers
194 views

hypersurface evolving with tangential velocity

If a hypersurface $S_t$ evolves with velocity only in the tangential direction, is $S_t \equiv S_0$ for all $S$? This is what I have read is true (or something very similar). Can someone give me an ...
2
votes
3answers
438 views

What's the motivation to add inner product and wedge product together in geometric product

I am reading some geometric algebra notes. They all started from some axioms. But I am still confused on the motivation to add inner product and wedge product together by defining $$ ab = a\cdot b + ...
4
votes
1answer
72 views

Explicitly writing out a differential 2-form

In Tu's An Introduction to Manifolds, one question asks: At each point $p\in \mathbb{R}^3$, define a bilinear function $\omega_p$ on $T_p(\mathbb{R}^3)$ by: ...