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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curvature$=0$ implies straight line?

The fundamental theory of differential geometry states that: If there is a given curvature $\bar{\kappa}(s)>0$ and torsion $\bar{\tau}(s)$ which both of them are differentiable and continuous in ...
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principal axis of a volume from moments of inertia

I'm trying to calculate the expression to find the principal axis of a volume via its moments. In the 2D case I can formulate the problem by expressing the moments around arbitrary axes $x' = x \cos ...
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28 views

$[D,D']$ where $D$ is a derivation and $D'$ is skew

This is a proposition in 33 page of Foundation in Differential Geometry - KN I need some detail. Let $D^r(M)$ be a set of $r$-form. Then derivation (resp. skew-derivation) of degree $k$ is a ...
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36 views

Component of a pushover vector by one-parameter transformation

I am curious about a step on the proof that shows Lie derivative of a vector field is equivalent Lie bracket. Following comes from Nakahara. We define integral curves by vector field X and Y as ...
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45 views

Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is ...
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323 views

Showing that two surfaces are not isometric/locally isometric

I am trying to solve an exercise which asks to show that two surfaces are not isometric and additionally that they are not locally isometric. The two surfaces presented are graphs. I know that if two ...
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37 views

Covariant Derivative to study Holonomy

Can someone provide an explicit definition to the $\mathbb{R}^{3}$-covariant derivative? For instance, I don't understand the following calculation. Let $$E_1 = (-\cos(u) \sin(v), - \sin(u) \sin(v), ...
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59 views

Smooth bijection has a dense open subset in which the inverse is also smooth

Let $f:\mathbb{R}^n \longrightarrow \mathbb{R}^n$ be smooth and bijective. Prove there exists open subset $U$ and $V$ dense in $\mathbb{R}^n$ such that $f: U \longrightarrow V$ has a smooth inverse. ...
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93 views

vector field as a projection of a constant vector field on an embedding space

(Also asked in math overflow. So if you dont find it appropriate here, moderators please remove this.) Suppose a tangent vector field is given on a planar curve and one asks the following question: ...
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31 views

How do I Prove that $M:=A(X_u)\cdot X_v=A(X_v)\cdot X_u$?

How can we show that $$M:=A(X_u)\cdot X_v=A(X_v)\cdot X_u\ ,$$ where $A$ is the shape operator and $X_u$, $X_v$ are the coordinate vectors?
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140 views

Lateral area of a cone with oblique base (elliptical base)

Can someone help me find the lateral area of a straight circular cone with oblique base ( inclined elliptical base) as seen at the following link. ...
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127 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
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361 views

How can we extend the proof that Lie groups are orientable to parallelizable manifolds?

Let $G$ be a Lie group with identity element $e$ and dimension $n$. I know we can take a non-vanishing $n$-form (call it $\omega$) on the vector space $T_eG$. For a Lie group, as we have an ...
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43 views

Integrating a vector field over curve in R^2 with differential forms

Sorry if this has been asked elsewhere; I know there are several questions on differential forms but I couldn't find the answer I am looking for. Imagine I have a vector field $F:\mathbb{R}^2 ...
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45 views

Scalar Curvature of a metric on the hemisphere, from a paper on the Min-Oo Conjecture

I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition: Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...
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54 views

Equation of tangent plane to a surface at certain point

So I am solving for an equation of a tangent plane to surface $z=x + \ln(2x+y)$ at the point $(-1,3,-1)$. I know I need to take partial derivative of that equation and plug in the point of the ...
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88 views

Principle Relative Curvature

Calculate the principal relative curvatures of the surface $y = x\tan\left(\frac{z}{a}\right)$. I have checked in my textbook, but the only definition of relative curvature I could find is $k = ...
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54 views

Show an immersion is locally one to one using the inverse function theorem

Using the inverse function theorem, show that an immersion is locally one to one. I am really struggling with this homework question can anyone give me a hint?
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106 views

globally defined function and restriction of a differential form

Consider the 1-form $a=p_1dx_1+p_2dx_2-H(p_1,p_2)dt$ defined on $R^5=(p_1,x_1,p_2,x_2,t)$ where $H$ is a globally defined smooth function that depends only on the coordinates $p_1$ and $p_2$. (a) ...
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59 views

Geometry of Curves and Surfaces

The following is a question regarding Elementary Geometry of Differentiable Curves. Venture a guess of the number of irregular values of $t$ for the trochoid with $h = 1, \lambda = \frac{m}{n}$, ...
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35 views

Extension of mean-curvature normal

Suppose $M$ is a two-dimensional manifold with metric $\bar{g}$, and $r: M \to \mathbb{R}^3$ is a (not necessarily isometric) embedding of $M$ into $\mathbb{R}^3$ with first fundamental form $g$ and ...
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40 views

Transversality of Subbundles

It is known that transversality of submanifolds is generic in the sense that two submanifolds could be made transversal by small perturbations. I was wondering if the same is true for subbundles of ...
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91 views

Sequence of critical values has no cluster point (Milnor, Morse Theory)

The following Claim is used in the proof of Theorem 3.5 in John Milnor's "Morse Theory": Claim: Let $f: M \rightarrow \mathbb{R}$ be a differentiable function on a manifold $M$ with no degenerate ...
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44 views

Is $\nabla_{X}:\Gamma(E) \to \Gamma(E)$ continuos aplication?

Be $\Gamma(E)$ smooth sections space of an vector bundle, $\Gamma(E)$ with Fréchet topology. Gived a conexion in E and a smooth vector field $X$, Is $\nabla_{X}: \Gamma(E) \to \Gamma(E)$ continuos ...
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What is the explanation of the equation 1-4.6 in the book “Applied Exterior Calculus”?

If the $n$ function $\{f^i(x^m)\}$ are of class $C^{\infty}$ in some neighborhood of $0$, then system of autonomous ODEs $$\frac{d\bar{x}^i(t)}{dt}=f^i(\bar{x}^m(t)),\quad\quad i = 1,\cdots, n$$ has a ...
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93 views

Geodesics of the Hyperbolic Plane.

Using the coordinates $\alpha=\log \frac{1+r}{1-r}$ and $\theta$ where $(r,\theta)$ are the usual polar coordinates, show that the segment of the y axis between $(0,0)$ and $(0,r)$ where $0<r<1$ ...
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45 views

Components of a Vector Field

This may be quite a petty question, but it's been bothering me for a while. So for the vector fields $X$ and $Y$, we can write them in component form as $X=X^{\mu}\frac{\partial}{\partial ...
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24 views

Dual Translation plane

An affine plane $\mathcal{A}$ is called a translation plane if the translation group of $\mathcal{A}$ operates transitively on the point set of $\mathcal{A}$. So how do we define the dual translation ...
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142 views

Upper-half space of a manifold with boundary

Suppose that $X$ is a manifold with boundary and suppose that $x$ is a boundary point. Define the upper half space $H_x(X)$ in $T_x(X)$ to be the image of $\mathbb{H^k}$ under $d \phi_0:\mathbb{R^k} ...
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185 views

Geodesic question

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability distributions on $\mathcal{X}=\{x_0,\dots,x_n\}$. Each $p=(p(x_0),\dots,p(x_n))\in ...
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59 views

Does this Condition Force this Differential 2-form to be 0?

Let $dw$ be a 2-form; $\alpha$ be a 1-form, $g$ a function (i.e., a 0-form), $X$ be a fixed vector field , all living in a 3-manifold, so that $$dw(X,.)=g\alpha .$$ Does this condition force $dw$ to ...
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63 views

Connection on $\operatorname{Spin}^\mathbb{C}$ spinor bundle

Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic manifold $(M, \omega)$, with a compatible $J$ and $g$, so \begin{equation} {W_ + } = {T^{0,0}}{M^*} \oplus ...
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36 views

basis theorem in holomorphic tangent space

I know that if $(x^1, \cdots, x^n)$ is a local coordinate system in a manifold $M$ then $\{\frac{\partial}{\partial x^1},\cdots, \frac{\partial}{\partial x^n}\}$ forms a basis for the tangent space ...
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92 views

Proof check for critical point definition with mean curvature

I'm currently trying to prove: "Definition: We say that a surface $S \in R^3$ is minimal if it is a critical point for the area functional" Starting with this: "If we consider a family of smooth ...
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39 views

Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
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Morse functions are dense in $C^{\infty}(M,\mathbb{R})$ questions.

Hi here is a proof inspired from the reference below. Feel free to get very technical with your comments so that at the end I understand it well. I am more concerned about the questions I added ...
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80 views

Manifold locally looks like a open set but not as a euclidean space?

I am reading about manifolds from the book by Millman. He says that manifolds locally looks like a open set, but there is no canonical way to make M look like a Euclidean Space, and so we can't define ...
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73 views

Does distance in hyperbolic space satisfy such properties which Euclidean distance have?

In Euclidean space $E^n$, the distance between two points $x, y$ is just $|x-y|$, and for each fixed $x_0$, the image $y\to\nabla_x|x_0-y|$ is $S^{n-1}$, so it satisfies ...
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45 views

How to compute saddle point index using sourcing flow lines?

Prove $Index_{p}(\bigtriangledown f)$= "dimension of sourcing flow lines from p" ,where p is a critical point. Attempt Near $ p \in Cr(f) $ in some coord. $ f(x) - f(p) = $ $\sum_j x_j^2 - \sum_k ...
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99 views

tubular neighborhoods - problem

Consider $\alpha:[0,L]\to \mathbb R^3$ a smooth regular simple closed curve, arc-length parametrized. Denote its trace by $\gamma$. Let $\epsilon >0$ be a real number and $N_\epsilon (\alpha)$ the ...
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181 views

Given Constant Ratio of Torsion to Curvature, Show Tangent times Constant Vector is Constant

Let $r(t)$ be a unit speed curve such that for all $t$, $\frac{\tau(t)}{\kappa(t)}=\cot(\theta)$ for some $0 < \theta < \pi$. Show that there is a constant vector $a$ satisfying $T(t) * a = ...
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45 views

PDEs along parametrized curves in $\mathbb R^2$

For Riemannian surfaces in $\mathbb R^n,n\geq 3$, the Laplace-Beltrami operator can be expressed in terms of the Riemannian metric and one can consider evolution equations (e.g. heat diffusion, wave ...
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107 views

Surface of revolution

This problem is from Dubrovin's Modern Geometry (Problem 8, Exercise 8.4). Show that the only surfaces of revolution with zero mean curvature are the plane and the catenoid (which is the surface ...
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Help modeling 3d vector field

I'd like some help in finding the correct mathematical description of the stuff below: A sheet is deformed by a mass on it (like in one of those pictures showing the effects of General Relativity). ...
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368 views

Chain rule with covariant derivative

Let $\mathcal{M}$ be a $n$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. Consider the following function: $$\tilde{F}(v) = \operatorname{d exp}^{-1}_{p} ...
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53 views

Boundaries of a cycloid

Supposing the Catenoid is parametrized by $x(u,v)=(acosh(u)cos(v),acosh(u)sin(v),au)$,where $u$ is a real number and $0<=R<2\pi$ and $a>0$ is fixed. Given the pair of parallel circles ...
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47 views

Show that x defines a coordinate neighborhood on the upper sheet of the circular cone C

Consider the map $x : U = (0, 2π) \times (0,\infty)\to{\mathbb R}^3$, $(\theta, v) \mapsto (v\cos\theta, v\sin\theta, v).$ a) Show that $x$ defines a coordinate neighborhood on the upper sheet ...
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Is the $\varepsilon$-neighbourhood theorem used in proving Homotopic transverse extension?

In Guillemin & Pollack page 71 I can't see where "For compact mfld Y,the map $\pi:Y^{\varepsilon}\to Y$ is a submersion" is used to show: "If for $f:M\to N$, closed subset $C\subset M$, closed ...
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93 views

Curvature tensor on the sphere

While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$ which is the ...
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73 views

Proving a curve of a logarithimic spiral is 1/s cot(theta)

A unit-speed plane curve $\gamma$ has the property that its tangent vector $t(s)$ makes a fixed angle $\theta$ with $\gamma(s)$ for all $s$. Show that: (i) If $\theta = 0$, then $\gamma$ is part of a ...