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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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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90 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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46 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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38 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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54 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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62 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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75 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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27 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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173 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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89 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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178 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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30 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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55 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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39 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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63 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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54 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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75 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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43 views

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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106 views

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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67 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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105 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 ...
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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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55 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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33 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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Last generalized curvature zero for a hyper-planar curve in $\mathbb{R}^n$?

Serret-Frenet frame has been generalized to curves in $\mathbb{R}^n$. I would like to enquire about what happens to the generalized curvatures if the curve belongs to one or more planes in ...
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82 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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38 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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97 views

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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66 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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68 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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41 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 ...