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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How to create a simple closed curve homotopic to the trefoil knot?

How to create a simple closed curve that is homotopic to the trefoil knot $\overrightarrow{\alpha} (t)= \left ( \left (3+ \cos (3t) \right) \cos (2t),\left (3+ \cos (3t) \right) \sin (2t),\sin (3t) ...
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28 views

Any elementary derivation of the Pfaff integrability condition?

Suppose in $\mathbf{R}^N$ we have a one-form field, $ \theta = \sum_{i=1}^N \theta_i d x_i $. The Pfaff integrability condition is $d \theta \wedge \theta = 0$. Is it possible to give an ...
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38 views

Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
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46 views

Stoke's theorem explanation

Firstly, Wikipedia informs me that Stoke's theorem might mean two different things so I'm pretty sure I'm talking about the general one. I'll just state what I understand from it and I'd like someone ...
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23 views

Citing a result on obstruction to Lagrangian Embedding

Let $L$ be a closed orientable Lagrangian embedding in $\mathbb C^n$. Then $\chi(L) = 0$, where $\chi(L)$ denotes the Euler characteristic of $L$. This fact is more or less stated in section 3.2 of ...
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30 views

arbitrary reparametrization

Let $\alpha: (a,b)\rightarrow \mathbb{R}^n$ of class $C^{\infty}$ with $\Vert\alpha^{\prime}\Vert>0 $ then if $\{ k,m,n\} \subset \mathbb{R}_+$ there repametrizacion $\beta: (m,n)\rightarrow ...
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26 views

Equivalence between pullback connections of smoothly homotopic maps

Let $f,g:M\rightarrow N$ be smooth maps between smooth manifolds such that there exist a smooth homotopy $H:M\times [0,1]\rightarrow N$ between them. If we have a principal bundle $P\rightarrow N$, we ...
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86 views

differential forms and orientation in Bott and Tu

I am reading Bott and Tu book on my own. If you have the text you can answer my question perhaps, if not it would be too long to explain it. I have gotten to page 29. I am confused about proposition ...
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58 views

All differentiable functions on $\mathbb{S}^n$

Consider the sphere $\mathbb{S}^n$ embedded in $\mathbb{R}^{n+1}$. Let $N$ be the north pole of the sphere and $S$ the south pole. Every point on $\mathbb{S}^n \backslash \{N,S\}$ is defined uniquely ...
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29 views

prove that ''pullback'' maps forms to forms

Suppose we have $A: V_{1}\to V_{2}$ where $V_{1},V_{2}$ are real vector spaces. Then $A^{\star}:\mathcal{J}^{k}(V_{2})\to \mathcal{J}^{k}(V_{1})$ where $\mathcal{J}(V):=\{\text{space of all ...
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18 views

Polynomial generator

If we let $\alpha$ be a multiindex, can we generate any polynomial in $\eta$ with coefficients as multiples of $\kappa$ $$ D_z^{\alpha}\text{exp}(i(\kappa(z)-\kappa(x)-\kappa'(x)(z-x))\eta)|_{z=x} $$ ...
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83 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
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45 views

integral of a 2-form over an oriented manifold

An old exam question: Let $M$ be oriented submanifold of $\mathbb{R}^{n}$ of dimension $k$, let $\omega$ be a $k$-form on $M$. 1) Define $\displaystyle\int_{M}\omega$ 2) let ...
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29 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
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37 views

Motivation for tensor density

Wiki has provided the basic definitions of the tensor density, but what I really want to know is the motivation and the advantage of this concept. Could anyone give me some ideas?
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53 views

Commutative differential graded algebra

The situation is the following: Let $V$ be a Lie algebra of finite dimension, say n, and let be the graded commutative algebra $(\bigwedge^{\bullet}V)^*:=(\bigoplus_{k=0}^n\bigwedge^kV)^*$ and define ...
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49 views

building a polytop from polytop and finding its volume

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ...
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48 views

Invariants of shape operators?

Let $S:V\longrightarrow V$ be a Linear Transformation, then the Characteristic polynomial of $S$ and therefore its Coefficient are invariant. Except the first and the last Coefficient that we know ...
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47 views

Differential Geometry and Origami

Would anyone know how to relate origami with differential geometry? I mean clearly you can see how geometry plays into it but how would you describe it in terms of differential
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32 views

Chain rule for tensor of family of tensor fields

Let $f_\tau$ be a $\mathbb R$-family (parameter $\tau$) of diffeomorphisms that map from $\mathbb R^4$ to $\mathbb R^4$. $f^*_\tau$ is the corresponding pullback (I think that is the correct term). ...
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61 views

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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74 views

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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26 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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37 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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159 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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35 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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51 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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30 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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81 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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110 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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154 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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36 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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35 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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29 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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53 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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37 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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62 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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53 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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32 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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36 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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68 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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43 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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n-Torus with antipodal points identified

If we have n-torus $S^1 \times S^1 \times S^1 \times ....$ n times, and $\mathbb{Z}_2$ acts on this just sending each component of $S^1 \times S^1 \times S^1 \times ....$ to its antipodal. What will ...
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63 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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23 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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102 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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177 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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43 views

When is a delta function a valid distribution?

If $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is nicely behaved, one can view $\delta(f)$ as a distribution (linear functional on $C^{\infty}_c(\mathbb{R}^k)$)- but what if you don't have nicely behaved ...