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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Vector equation and curvature

Vector equation $r(t)=2\cos(t)\mathbf i + 3\sin(t)\mathbf j\ \ (0 \le t \le 2\pi)$ represents ellipse. I need to find curvature of this ellipse on endpoints of x and y axis that are given with ...
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95 views

The lambda invariant

Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane. ...
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64 views

derivatives of coordinates on a riemann surface

Let $X$ be a compact connected Riemann surface of genus $g>0$. Let $(\omega_1,\ldots,\omega_g)$ be a basis for $H^0(X,\Omega^1)$. Let $x$ be a point in $X$ and let $z:U\longrightarrow B(0,1)$ be a ...
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173 views

Is the extra condition in this definition superfluous?

I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
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114 views

Decomposition of linear partial differential operators

I was wondering about the following: Let $M$ be a smooth, second-countable (possibly noncompact) manifold and let $E$ and $F$ be smooth vector bundles over $M$. Can every smooth linear partial ...
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859 views

How do we prove that a sphere maximizes the volume enclosed among all simple closed surfaces of given surface area?

How do we prove that among all closed surfaces with a given surface area, the sphere is the one that encloses the largest volume, and not do it by cases? so far I've tried is that I know the formula ...
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302 views

Question on Stokes' Theorem

Suppose $M$ is a smooth manifold and $f$ is a real valued smooth function on $M$. Set $N:=f^{-1}([0,1])$ and suppose $N$ is a compact submanifold of $M$. Let $\mu$ be a volume form on $M$ and $v$ a ...
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365 views

a question about the geodesic circle

How to show that the geodesic circles have constant geodesic curvature on a surface of constant curvature? Thanks
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166 views

Are Lie Groups Homogeneous Spaces?

Is any Lie Group a homogeneous space?
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153 views

costructing a diffeomorphism

Let $i:N\to M$ be a smooth embedding, $\pi:E\to N$ a vector bundle and $s_0:N\to E$ is its zero section. I have an open neighborhood $U$ of $s_0(N)$ in $E$, and $f:U\to M$ is a smooth map such that ...
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144 views

A particular pulling back and lifting of metric

Let $\Sigma$ be a $n-1$ dimensional space-like submanifold of a $n+1$ dimensional space-time $(V,g)$ and let $x \in \Sigma$. Then $(T_x \Sigma)^\perp$ is of dimension $2$ and is time-like. Such a ...
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253 views

A particular method of pulling back a metric on a submanifold

Let $S$ be a $(n-1)$-submanifold of a $n$-manifold $M$ and that be a submanifold of $(n+1)$-manifold V. (All of the manifolds are assumed orientable) Let $V$ carry a metric of signature $(1,n)$. Using ...
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16 views

Finding parallels surface of revolution

$$X(u,v)=(f(u)\cos(v),f(u)\sin(v),g(u))$$ where $v$ is the rotation angle around $z$ axis. What is the parallel of this surface of revolution?
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22 views

Differential Geometry concept: fit the X, Y, Z data and express X=F(Y,Z)

I've been given a project to fit the X, Y, Z data and express X=F(Y,Z); then compute the principle curvatures at any point on the surface (or at least the interior grid points). I know this has ...
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19 views

Every point in the codomain is a regular point

Let $0<r<1$ and define $f:\mathbb R^3\to\mathbb R$ by $$f(x,y,z)=(x^2+y^2+r^2-z^2-1)^2-4(x^2+y^2)(r^2-z^2).$$ Let's denote $x^2+y^2=a,r^2-z^2=b$. I don't know if this is allowed, but I just ...
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24 views

Completion of hyperbolic 3-manifold (Thurston's lecture note)

I need help with a part of Thurston's lecture note. p.55~56 in the 4th lecture note states that when obtaining hyperbolic manifold by gluing polyhedra having some ideal vertices, the set of ...
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22 views

Reparametrization of curve

I found this in a book: Let $\alpha(t) = (g(t),h(t),0)$ be a regular curve in the $xy$-plane. If $g'(t) \neq 0$ for all $t$, then $g$ is strictly increasing. Thus, it is one-to one and has an inverse ...
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19 views

Definition of roulette (curve)

I want to give a formal mathematical definition of a roulette, a curve described by a point attached to a given curve as that curve rolls without slipping, along a second given curve that is fixed. I ...
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14 views

Path of constant relative gradient to a cone.

The equiangular or logarithmic spiral has the property that the angle between any tangent and the radial line is a constant. I am looking for a curve with the same property with respect to a conical ...
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21 views

Explicit formula for the (n-2)th derivative of the Jacobi equation

The $n-2$ order derivative of the Jacobi equation is given by: $$\frac{D^n}{dt^n} V_i+\sum\limits_{l=0}^{n-2} \binom{n}{k} (\nabla_{\gamma '}^{(n-2-l)}R)(\gamma ' ,\nabla_{\gamma '}^{(l)} V_i)\gamma ...
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Proving that $\Phi_{t*}[\mathbb{Y},\mathbb{Z}]=[\Phi_{t*}\mathbb{Y},\Phi_{t*}\mathbb{Z}]$

Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$. Let $\Phi_t$ denote the flow of $\mathbb{X}$. You are given that $\displaystyle L^j_{\mathbb{X}}[\mathbb{Y},\mathbb{Z}]=\sum_{k=0}^{j} ...
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14 views

Another Problem integrating when attempting a solution with the Poincaré Lemma

a) I think that the answer should be $d\nu=10z dx \wedge dy \wedge dz$ b) and c) are easy. d) This is part I am having troubles with. $\begin{align} i_{\hat{\mathbb{X}}_t}\beta &= ...
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35 views

Pull-back of a one-form on a sphere.

Let $\iota: S^2 \to \mathbb{R}^3$ be the inclusion map and choose a chart $(U,f)$ on $S^2$, where $U=\{(x,y,z)\in \mathbb{R}^3: z>0\}$ and $$f: U \to \mathbb{R}^2,$$ $$ (x,y,z)\mapsto (x,y). $$ I ...
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52 views

What geometric shape is “the perfect milkshake container”?

A sphere is the best shape for a snowball if you want to maximize the amount of time before the snowball melts. This is because the ratio of the surface area divided by the volume is the smallest. ...
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11 views

Surfaces swept out by trihedron vectors

Surfaces swept out by unit tangent of a curve on a surface is developable. Are normal and bi-normal swept out surfaces also developable?
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17 views

Variation of geodesic and Jacobi field

I was reading on Jacobi field of a geodesic, and noticed that given a geodesic $\gamma$ it is defined using the term of variation or family of geodesics $\gamma_s$ but never mentioned how to create ...
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27 views

Geodesic loops in Riemann homogeneous spaces

Let M be a Riemannian homogeneous space, i.e. the isometry group acts transitively. Prove: any geodesic loop (with possible angle at the starting point) is a closed geodesic (smooth at the starting ...
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19 views

Oval surface with $ K \ge 1 $ which is unit sphere

Let $ S$ oval surface with $ K \ge 1$ . If there is exist an open unit sphere interior of $ S$, then $S$ is unit sphere...Can anyone give me an idea of the solution...thanks in advance...
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19 views

Define the concept of the Shape Operator and Fundamental Forms

I am confused about the relationship between three concepts: shape operator, first fundamental form, second fundamental form. I would like someone to provide me with a basic definition of these terms ...
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43 views

About the existence of a certain form

Let $M$ and $N$ be compact and oriented manifolds, with empty boundary. Let $n=\dim(N)<m=dim(M)$ and let $i:N\rightarrow M$ be an embedding. I am asked to prove that there exists some ...
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55 views

Exactness of a certain sequence

Let $M$ be a connected manifold of dimension $m$ and let $\beta\in H^{1}(S^{1})$ such that $\int_{S^{1}}\beta = 1$. Let $\pi_{1}:M\times S^{1}\rightarrow M$ and $\pi_{2}: M\times S^{1}\rightarrow ...
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22 views

Euler Lagrange of a Curve

Let $C(s) = (x (s), y(s))$ be a closed curve inside a plane where $s$ is the parametric arc length parameter. What is Euler Lagrange equation for the following functional $$-\int_0^L \nabla C ds$$ ...
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26 views

Weierstrass-Enneper representation formula

State and prove the Weierstrass-Enneper representation formula. I have tried to find in some books but failed. I will be thankful if some one help me out of this.
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21 views

Change of variables and relations between partial derivatives.

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a map given by $f(s, t) = (s, st)$. Do we have the following identity \begin{align} \left( \begin{matrix} \frac{\partial}{\partial s} \\ ...
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25 views

Natural curvature tightening of parametric curve

I'm looking to compute the "tightening" of curvature for a curve (mine is mainly 2D but could be of any dimension). In particular, since I am mainly 2D, I'm staying away from the cross-product based ...
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25 views

Prove that the torus is a regular subvariety of $\mathbb{R}^3$

Consider the inclusion of the torus in $\mathbb{R}^3$, prove that the torus is a regular subvariety. I have the next idea. If we describe de points of the torus as pairs $(u,v)$. We have to prove ...
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16 views

Possibly notation problems involving Integration and pullbacks on k-forms

$^*$ means the pullback of a k-form in this example. I cannot see how the underlined expressions have been found 1) I think that $(c \circ G)^*\omega = G^*(c^* \omega)$ but I cannot see why $c^* ...
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20 views

fibration sequence of projective spaces

Question~1: How to construct a fibration sequence $$ S^3\to S^2 \to \mathbb{C}P^\infty\to \mathbb{H}P^\infty ? $$ Does $$S^3\simeq \Omega \mathbb{H}P^\infty ? $$ (Since $\mathbb{C}P^\infty\simeq ...
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30 views

Gaussian curvature proof

I can show the first part but not sure how to proceed after that.
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14 views

Surjective $\gamma \colon I \to M^1$, $\gamma (t_1)=\gamma (t_2)$ can be extended to a periodic parametrization of $M^1$

Suppose that $\gamma \colon I \to M^1$ is a smooth surjective curve in a Riemannian connected 1-dimensional manifold. Furthermore, suppose that it is parametrized via arc lenght i.e.:$$||\dot ...
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20 views

Symmetry of Christoffel symbols of the second kind

I was reading the article: http://physicspages.com/2013/12/22/christoffel-symbols-symmetry/, and I do not understand this: In the locally flat frame, this equation reduces to $\displaystyle ...
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37 views

Stuck in Preissmann's theorem

I am stuck on following the proof of Preissmann's theorem, whose statement is that Let $(M,g)$ be a closed connected Riemannian manifold of negative sectional curvature. Then every nontrivial ...
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25 views

Find the lie derivative of a particular integral

Let $\mathbb{X}=(1,y)$ be a vector field on $\mathbb{R}^2$. Let $\Phi_t$ be the flow of $\mathbb{X}$. The flow of $\mathbb{X}$ I have calculated to be $\Phi_t(x,y)=(x+t,ye^t)$ Given a function ...
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50 views

Application Question - American universities strong in Differential Geometry?

Can anyone recommend some American universities (except those top 10 ones such as Harvard, Princeton, SUNY and Umichgan etc. ) which have departments with a solid focus on Geometry and Topology, ...
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17 views

Poisson bivector on the product of two manifolds

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. A Poisson bivector on $X$ is defined by \begin{align} \pi_X = \sum_{i,j} ...
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13 views

If the linearization of a section is surjective on a slice, is the image under a submersion also a smooth manifold?

Let $V \rightarrow M \times N_1 $ be a vector bundle, where $M$ and $N_1$ are smooth manifolds and $s: M \times N_1 \rightarrow V$ a smooth section such that if $s(p,q) =0$ then $$ \nabla ...
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45 views

How to define an action of vector field on $C^{\infty}(M)$?

Let $M$ be a manifold. Let $\hat{X}$ be a vector field on $M$. How to define an action of $\hat{X}$ on $C^{\infty}(M)$? Thank you very much.
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15 views

alternate definition of winding number?

If c is a singular $1$-cube in $R^2-\{0\}$ with $c(0)=c(1)$ , show that there is an integer $n$ such that $c-c_{1,n}=\partial c^2$ for some $2$-chain $c^2$. Here $c_{R,n}=(R\cos 2\pi nt,R \sin ...
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20 views

Implicit partial second derivatives from coupled equations

I have three functions defined in two variables. $f(x,y)$ $g(x,y)$ $h(x,y)$ I wish to find the partial derivatives $f_{gg}$, $f_{hh}$, and $f_{gh}$ and evaluate them at a particular point. In this ...