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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Use of Implicit Function Theorem to provide examples of Manifolds

Suppose we know Definition 1 and Theorem 1, and want to prove Theorem 2 given below. Definition 1: A subset $M$ of $R^n$ is called an k-dimensional manifold (in $R^n$) if for each point $x\in M$ the ...
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229 views

Correct use of substitution rule for Integration on Riemannian manifolds

Let $(N,g_N)$ be a Riemannian manifold and let $\psi: M \rightarrow N$ be a a diffeomorphism. Now I know how the Riemannian metric on $M$ defined by the pull-back of the metric on $N$ looks like (this ...
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Are there more types of differential in mathematics?

I am familiar with two types of differential normal differential: $$d^2x^a$$ covariant differential: $${\mathcal D^2x^a}=d^2x^a+\Gamma^a_{bc}dx^bdx^c$$ (where the covariant differential is broken ...
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69 views

Non-Linear Beltrami Vector fields

Consider two concentric toruses, and let $\Sigma$ be the domain interior to the greater torus and exterior to the smaller torus. Is it possible to find a vector field $\mathbf{b}$ satisfying the ...
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116 views

Can 24 lines on a cubic surface be realized as 24 identical spiral rods?

It's possible to put 24 lines on a cubic surface. 27 lines is possible, but I don't have a great picture for that surface. It turns out that the 24 lines can be built with Zome. I'm thinking that ...
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69 views

Convex polyhedral decomposition of spheres

Is there a decomposition of $S^2$ into $k$ (geodesically) convex polyhedra that are congruent to each other? What about $S^n$ for $n>1$? Remarks: A polyhedron is defined as an area enclosed by a ...
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Mean curvature operator in S^n

Consider the sphere $S^n$. By using the stereographic projection we can identify $S^n \setminus N$ with $\mathbb{R}^n$, where $N$ is the North pole of $S^n$. The metric then is given by $\frac{dx^2}{(...
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Convexity of a function

Suppose we have $F: R^n \longrightarrow R$ , $P: R^n \longrightarrow R^n$ and $G: R^n \longrightarrow R$ all nice- let's say given by polynomial and $P$ is invertible - such that $F(x) =G( P(x) )$. ...
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82 views

Pushforward/derivative of map between surfaces in $\mathbb{R}^n$

If $f:M \to N$ is a smooth map between compact closed hypersurfaces $M$, $N \subset \mathbb{R}^n$, does it make sense to write the pushforward as $Df$, the total derivative? Because usually we require ...
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67 views

How to prove an isotopy relative to a point exist?

Let $M$ $ $ be a differential manifold, and $f$ a diffeomorphism on $M$ which is isotopic to $id$. Assuming that $x\in M$ is a fixed point of $f$ and the orbit of $x$ under the isotopy is a trivial ...
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356 views

Constant rank map

Suppose $F:M\to N$, $M,N$ smooth manifolds and $M$ connected. I proved that for each $p$ in $M$ there exists smooth charts near $p$ and $F(p)$ in which the coordinate rapresentation of $F$ is linear. ...
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315 views

Connections and curvature on general fiber bundles

Let $\pi: E \to M$ be a fiber bundle with fiber $F$. We can define a connection to be a projection $v: TE \to VE$ satisfying $v \circ v= \text{Id}$, where $TE$ is the tangent bundle of $E$ and $VE$ is ...
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80 views

About Thom theorem (representation submanifold for $H_{n-2}(M)$)

Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold. And in the Harper and ...
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51 views

List of minimal surfaces embedded in the 3-sphere

Is the set of possible areas of closed, embedded, minimal surfaces of the 3-sphere discrete?
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874 views

Top deRham cohomology group of a compact orientable manifold is 1-dimensional

Let $M$ be a compact smooth orientable manifold of dimension $n$. I am looking for a simple proof that $H_{dR}^n(M) \cong \mathbb R$. Equivalently, an $n$-form which integrates to 0 is exact. I can ...
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323 views

Plane curve: orhogonal projection and closest points

I am reading 'Differential Geometry of curves and surfaces' by Banchoff and Lovett and I'm confused about the following statement on page 28: "Let two curves $C_1$ and $C_2$ have a regular point $P$ ...
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61 views

Prove that $X$ parametrizes a regular surface $M$ in $\mathbb{R}^3$ and determine for which values $p$ the curve $y$ is geodesic on $M$.

Real valued functions $f,g:\mathbb{R}_+ \rightarrow \mathbb{R}$ are $f(u) = e^{-u}$ and $g(u) = \int \sqrt{(1-e^{-2t}} dt$. Given $X:\mathbb{R}_+ \rightarrow \mathbb{R}^2$ with $X(u,v) = [f(u) \cos v,...
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Complex structure on the product of two complex Kähler manifolds

I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kähler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think ...
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Isomorphism between tensor product and set of all tensors

I'm new to tensor and quite confused. First of all, could anyone provide any friendly reference about tensors which explains the idea behind those boring-looking definition? Then is my problem. Let ...
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119 views

Twisted tori: discrete and continuous

Taking the advice of Mariano Suárez-Alvarez, I moved this question from MO to MSE: Motivation Let me introduce twisted (discrete) tori: Consider the Cartesian graph product $\mathcal{C}_n = C_n \...
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94 views

Is there an action functional whose critical points are the geodesics of a arbitrary connection on TM?

Geodesics of the Levi-Civita connection may be defined as the critical points of the action functional $S[\gamma]=\int \lvert\dot{\gamma}\rvert\,dt$ (or square it, if you like). The Euler-Lagrange ...
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How to apply Gauss's theorem when the metric is unknown

Let $f:U \to \mathbb{R}^3$ be a surface, where $U=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|<3, |u^2|<3\}.$ Consider the two closed square regions $F_1=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq1, |u^2|\leq1\}...
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Hopf theorem for non-orientable manifold

I have an exercise which says that :extend the Hopf-Poincare theorem for non-orientable manifold with the indication using the double covering. I have got stuck for long time, so I don't know if ...
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192 views

A question about a definition of Ricci curvature

Let me first quote the definition of Ricci curvature from Wikipedia. Let $M$ be an n-dimensional Riemannian manifold equipped with its Levi-Civita connection $\nabla$. The Riemannian curvature tensor $...
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59 views

Reconstructing paths on the sphere from the ratio of acceleration to velocity

Given a path $\gamma:[0,1]\to \mathbb C$, we can determine $\gamma$ from information about its derivatives. For example, $\gamma$ is determined by $\gamma(0), \gamma'(0)$, and $\gamma''(t)$. This ...
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180 views

Conjugate function reverse angle in a neighborhood

Can you please help me with this question. Prove that if conjugate of $F$ is analytic in a neighborhood of $z_0$ in $C$ then $F$ reverses angles at $z_0$. Can anyone please explain that for me? what ...
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94 views

laplacian for functions problem with the integral on manifolds

I'm following the proof of the local expression for the Laplacian on a compact manifold and I'm having problems understanding how the integral on a manifold translates into an integral in $R^n$, in ...
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200 views

Homogeneity lemma

I am studying Homegeneity lemma. I am not understanding the following paragraph: Given any fixed unit vector $c \in S^n$, consider the differential equations $\frac{dx_i}{dt} = c f(x_1,x_2,\ldots,...
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When can we write evolving curves as curves over a fixed curve?

Suppose $\gamma(t)$ for each $t$ is a curve. We may write $$\gamma(t)(s) = \gamma_0(s) + d(t,s)N(s)$$ where $\gamma_0$ is some fixed curve, $N$ is the unit normal vector and $d$ is a distance from $\...
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Calculation of |[X,Y]^V|

I want to follow the proof of Theorem 3.1 in "On Eschenburg's Habilitation on Biquotients" - Wolfgang Ziller. The situation is as follows: $Q$ is a biinvariant metric on $G$. So from the ...
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243 views

Closed Geodesics as minimisers of action functional

Suppose I have a Riemannian surface $(M,g)$. It's clear that closed geodesics are critical points of the length functional $l(\gamma)=\int\left|\gamma(t)^{\prime}\right|_{g(\gamma(t))}dt$ over the ...
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Unit partition to produce smooth function from continuous ones

Given a positive continuous function (except on closed set, where is zero ) on a smooth manifold how to find a smooth function under the same conditions being less (or equal) than this one continuous?...
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A series of Lemmas about $C^{\infty}$ vector fields

Warner page-37: Theorem: Let $X$ be a $C^{\infty}$ vector field on a differentiable manifold $M$ For each $m\in M$ there exist $a(m)$ and $b(m)$ in extended real line, and smooth curve $$\gamma_m:(a(m)...
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flows on a manifold and liebracket

I have the following question: Let $M$ be a smooth manifold and let $p \in M$. Furthermore let $X$ and $Y$ be two vector fields in a neighbourhood $U$ of $p$ and consider their flows $\varphi^{X}(t,x)$...
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148 views

Smooth deformation retracts

Under what circumstances can it be concluded that if two items from the smooth category are related by a topological relationship, then they are also smoothly related in the corresponding way? For ...
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On what quantities is a (differential geometric) connection not allowed to depend?

If we parallel transport a vector $v$ along a closed curve $c_1$, the vector will end up as $v'=A_{c_1}v$, where $A_{c_1}$ is an element of the holonomy group. If we consider composition of curves ...
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337 views

De Rham cohomology of the euclidean space without n lines

How can I compute the de Rham cohomology of $\mathbb{R}^3$ minus n lines through the origin? I would like to do it with the Mayer-Vietoris sequence (which is the only thing I know to calculate ...
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181 views

Isometry from a surface to itself.

We are talking about surfaces in $\mathbb{R}^3$ here. I know that not every isometry from a surface to another is a congruence. But what about isometries from a surface to itself? Can someone give ...
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Different definitions of the complex Grassmannian

I have come across two different definitions of what I suspect is the same object. Both are called the complex Grassmannian: 1: $U(n)/U(k)\times U(n-k)$ 2: $SU(n)/(S(U(k)\times U(n-k))$ What is the ...
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Help with this geometric PDE weak formulation and solution

Suppose $X:A \subset \mathbb{R}^n \to \mathbb{R}^{n+1}$ is a parametrisation of a surface $\Gamma$ such that $X(\theta_1, \theta_2) = (\theta_1, \theta_2, h(\theta_1, \theta_2))$ (i.e., $\Gamma$ is a ...
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Different proofs of support theorem for Radon Transform

Let $\mathcal{H}$ be a set of all hyperplanes in $\mathbb{R}^n$. Radon transform of function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ is defined as function $R[f] \colon \mathcal{H} \to \mathbb{R}$ ...
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Reversing a roulette on a straight line - solving for a parameterization?

(See below for update.) I would like to reverse the following equations. Here, the path is traced out by the polar origin of the wheel. You can put the trace point anywhere on or off of the wheel ...
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Deriving PDE to evolve a planar curve to a circle, while preserving its length.

I would like to ask, how could I derive a PDE that evolves a planar curve to a circle while preserving its length ? (The problem states that there is no need to show that the steady state is a circle, ...
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131 views

Determinant of the Laplacian of a surface is this correct?

given a surface with metric $ g_{ab} $ i would like to evaluate the functional determinant of the Laplacian in the form $ - \partial _{s} \zeta (0,E^{2})=\log\det( \Delta + E^{2}) $ then i need to ...
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A lower positive bound on the number of closed orbits with given energy for a mechanical system

Let be given a mechanical system with configuration manifold $M,$ potential energy $V$ and kinetic energy $K$ corresponding to a riemannian metric on $M.$ Its dynamics is determined by the Euler-...
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Unified Definition of a Surface

As I noted in this question, there's a lot of inconsistent terminology in use with regard to "curves", "smooth curves" etc and similar comments could apply to the definition of a surface. To get all ...
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143 views

A question on generalized Gauss-Bonnet theorem

I am trying to learn about basic characteristic classes and Generalized Gauss-Bonnet Theorem, and my main reference at the moment is From Calculus to Cohomology by Madsen & Tornehave. I know the ...
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200 views

How does a geodesic equation on an n-manifold deal with singularities?

My general premise is that I want to investigate the transformations between two distinct sets of vertices on n-dimensional manifolds by: Minimalizing the change in the fundamental shape of the ...
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44 views

Evaluating the “regularity” of a mapping $\mathbb{R}^2 \rightarrow \mathbb{R}^2$

Let $R \subset \mathbb{R}^2$ and $R' \subset \mathbb{R}^2$ be two regions in the plane, and $F: R \rightarrow R'$ a smooth map. I would like to find a reasonable measure of the "regularity" (...
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301 views

How to construct a vector field without zero on an open manifold?

a friend asked me to pose the following problem: It is known that on an open manifold (connected, not compact and without boundary) there exists a vector field without zero, since its Euler ...