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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Topology of “line integral convergence” on the space of curves

Let $C^1(I,\mathbb{R}^n)$ be the space of $C^1$ curves. Give it the topology that satisfies that convergence of a sequence of curves $\gamma_n \to \gamma$ occurs iff these conditions hold: a. ...
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Finding curves with special constraints. How do i verify the answer?

Let $O$ be a fixed point on the plane. Find all curves $\gamma : \mathbb{R} \to \mathbb{R}^2$ such that if $C(t_0)$ is the centre of the curvature of $ \gamma$ at $t_0$ then the angle $\widehat{ ...
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integration of differences of two inverse functions of CDF

When F, G are both cumulative distribution functions, $\int_0^1 | F^{-1}(y) - G^{-1}(y)|dy = \int_{-\infty}^\infty |F(x) - G(x)|dx.$ The paper (Vallender 74') says that "The equation follows from ...
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Prove using Math (Multivariable calc. or Diff. geometry proof) that $(x_u, y_u, z_u)\times(x_v; y_v; z_v)$ for a parametrization $X$ of $S$.

For a parametric surface $x = (x(u, v); y(u, v); z(u, v));$ the derivatives $x_u$ and $x_v$ are vectors in the tangent plane. Thus, their cross product = $(x_u, y_u, z_u)\times(x_v; y_v; z_v)$ is ...
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show that if $M \times \mathbb{R}^{n}$ is orientable than so is $M$

I need to show if $M \times \mathbb{R}^{n}$ is orientable than so is $M$, where $M$ is connected manifold. $R^{n}$ has standard orientation (determined by standard basis ) and by the assumption $M ...
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39 views

Fundamental cycle $[Y]\in H_{2n-2}(X_{\mathbb{R}},\mathbb{Z})$ of irreducible analytic hypersurface on a complex manifold X.

I am studying about complex manifolds and I am trying to understand the following statement. Let $Y\subset X$ be an irreducible analytic hypersurface, where $X$ is an n-dimensional complex compact ...
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25 views

Construct two-form

I should give an example and construct a two-form on the 2D sphere. I know how to construct one-form on the 2D sphere, but I have no idea how to continue with the two-form.
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36 views

Finding boundary coordinate chart

I need to calculate the boundary coordinate chart of the manifold with boundary $$M=\{(x,y,z)\colon x^2+y^2+z^2=1, z\ge0\}$$ If I define $U=\{ (u,v)\colon u^2+v^2\lt1 \}$ and ...
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Doubt in an expression of the vector field

I know that $$\{\dfrac{\partial}{\partial x_i}:i=1(1)n\}$$ is a basis of the $n$ dimensional tangent space $T_p(M)$ [Vector space over $\mathbb R$] at the point $p$ on $M.$ Again I came to know ...
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31 views

Two cylinders cutting

Find equation of non-circular cylinder $ f (y,z)=0 $ which cuts a circular cylinder $ x^2+y^2 = a^2 $ to produce an intersection curve of constant geodesic curvature on the circular cylinder. What is ...
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Convolution of functions defined on manifold

Let $M$ be a Riemannian manifold with fixed volume form $\mu$. How to define a convolution of two 'functions' $f,g \in L^1(M)$? I will be grateful for an answer or for giving me some refrence where it ...
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39 views

Estimating vector fields on the product of compact manifolds

Let $M,N\subset \mathbb{R^n}$ be compact embedded manifolds, $X_1,...X_i$ vector fields on $M\times N$ and $\delta\colon M\times N\rightarrow (0,\infty)$ a continuous function. Are there vector ...
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47 views

Flow past a moving sphere

When the air passes over a moving sphere the boundary layer separates opposite to the direction of travel. The separation occurs at different positions to the back of the moving sphere. If separation ...
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37 views

Vector fields and solutions to their ode

The problem is stated as follows: Suppose given a vector field $v$ on $\mathbb{R}^n$ defining a linear system, so $v$ is described by a linear map $\mathbb{R}^n \mapsto \mathbb{R}^n$. Also give ...
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9 views

How to find the points on an Ellipsoid such that the normal has equal angles with the coord. axis?

I have seen that one could find the points of an Ellipsoid: $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$$ But i can see the way to reach them.
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29 views

Gauge transformation laws, proof in Kobayashi & Nomizu Foundations of Differential geometry

I have two questions about this proof found in K&N's Foundations of Differential Geometry. 1) Can someone please explain how they deduce ...
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30 views

Quasilinear PDE

I begin with a PDE involving a function $u=f(x,y)$. The PDE is in the form $$f(x,y,u)u_x + g(x,y,u)u_y = h(x,y,u)$$ I try to solve the following system of odes involving the characteristic curves ...
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34 views

Tangential divergence

While reading a textbook i came across a concept of tangential divergence, I would like to get the proof or the idea behind it . It is defined as follows: For a smooth vector field $v: \partial ...
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36 views

Mean curvature of polar parametric surface

For the purposes of modelling a fluid mechanics experiment, I'm dealing with a convex surface parametrized by the azimuth $\theta$ and an arc length $s$ along the surface. The points on the surface ...
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33 views

Understanding Euler density

I know the definition of Euler density in terms of antisymetrized contractions of products of the Riemann curvature tensor, ie Euler density is the $\mathcal{R}^n$ in these formulae: And I know ...
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35 views

Differential Geometry-Adjoint of Exterior derivative

How to prove $$\delta=(-1)^{n(p+1)+1}*d*$$ if $\delta$ is the adjoint of exterior derivative d and * is Hodge Star
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Curve traced by following a Jordan curve.

Let $C$ be a Jordan curve in the plane and fix a point $x$ on $C$. For a point $v$ on $C$, define the curve $C_{v_a}$ as follows: draw a line from $x$ to $v$ and two additional lines to a point $a$ so ...
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74 views

Can one define wedge products using determinants for $n$-forms?

I was talking to Ted Shifrin in math chat yesterday and he mentioned there is a way to define wedge products using determinants. As far as I understand, given a set of vectors $x,y,z,v,u... \in ...
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the laplacian in real coordinates and complex coordinates

In the notes by Szekely it says on page 35 that since $\nabla_k\frac{\partial}{\partial \overline z_l}=0$ it follows in holomorphic coordinates that the Laplacian is $\Delta f=g^{j\overline ...
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Are the following sets a submanifold?

I have got stuck in showing whether this particular subsets are submanifolds(smooth): Let $f:\mathbb{R}\to\mathbb{R}$ be the function $f(x)=|x|$. Then the graph of the function f is smooth ...
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25 views

Adjacent angle Theorem in planes of constant curvature - easy geometric proof?

I am trying to proof the Adjacent angle Theorem in planes of constant curvature (2-Sphere, euclidean plane, hyperbolic plane) i.e given 4 points $a,b,c,d$ such that $d$ is lying on a shortest curve ...
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71 views

Curvature and Pfaffian forms in terms of the Riemann tensor

I am teaching my self differential geometry, but I am mainly familiar with classic tensor notation. In modern Cartan exterior form notation the curvature forn $\Omega$ and the Pfaffian seem to do the ...
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35 views

Moment map and Hamiltonian

Take the manifold $M$ to be $M=\mathbb{R}^6=\mathbb{R}^3\times\mathbb{R}^3$ (hence $x\in M$ is given by $x=(p,q)$ with $p$ and $q$ three dimensional vectors) and take the possion bracket on $M$ given ...
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Svarc-Milnor Lemma to prove that finite index subgroup of f.g. group is finitely generated

I found a proof using Švarc-Milnor lemma (the Lemma is prop. 1.19 here) of the well known fact that a subgroup of finite index of a finitely generated group is finitely generated (here a proof with ...
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51 views

Manifold and the topology of $\mathbb{R}^{n}$

A manifold $M$ is defined in particular as being locally homeomorphic to $\mathbb{R}^{n}$. Homeomorphisms can be defined in terms of how they map open sets, namely an homemorphism $f$ and its inverse ...
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Question about uniform wire and its application to find centroid

A uniform wire has the shape of that portion of the curve of intersection of the two surfaces x^2+y^2=z^2 and y^2=x connecting the points (0.0.0) and (1.1.square root 2) Find the z-coordinate of its ...
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Application of Bonnet's theorem for lines on a surface

Curvature of any line on a $\mathbb R ^2$surface has two orthogonally resolved components normal and geodesic curvatures $ \kappa_n, \kappa_g $, with $$ \tan \gamma = \dfrac{\kappa_g} {\kappa_n} $$ ...
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A question on diffeomorphisms and their relation to active coordinate transformations

I've been reading Sean Carroll's notes on General Relativity, http://arxiv.org/pdf/gr-qc/9712019.pdf . I've got to chapter 5 (page 133) and am reading the section on diffeomorphisms in which Sean ...
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33 views

Regular Surface Patches

Which of the following are regular surface patches? Let $u,v\in \mathbb{R}$. $(i)$ $σ(u, v)=(u, v, uv)$ $(ii)$ $σ(u, v)=(u, v^2, v^3)$ $(iii)$ $σ(u, v)=(u + u^2, v, v^2)$ I'm not sure how to show ...
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31 views

The Jacobian of the exponential along a geodesic

I am reading a paper that uses but does not define the following concept: what is understood by "the Jacobian of the exponential map along a geodesic (beetween two points)"? Is this only defined for ...
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47 views

Lie bracket equation

I need to solve the equation system of the Lie brackets of vector fields. So I want to find vector fields $X,Y,Z$ such that $F:(\mathbb R^3,\times)\to (V(\mathbb R^3),[.,.])$ , ...
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Toroidal Fibration

I am looking for a definition of toroidal fibration. In particular I am interested in a toroidal fibration of spheres and in finding the conditions under which they exist.
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29 views

Left invariant vector field question

This question is an exercise from Jeff Lee's Manifolds and Differential Geometry Let $\tilde{X}$ be the left invariant vector field corresponding to $X\in L(V,V)$ (That is, $\tilde{X}_g=(dL_g)_e X$). ...
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23 views

How to find maps between circles on $\mathbb{R}^3$?

In my notes there is a proposed hard exercise asking to find maps on the set of circles on $\mathbb{R}^3$. I cannot understand what exactly this even means. The example exercise, previous to this one, ...
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51 views

Natural morphism of sheaves $pr_1^{-1} F \otimes pr_2^{-1}G \rightarrow j_*(F\otimes G)$

I am reading a book, and the book said there is a natural map, which I don't know how. Can someone help me please? I can try to define the map, but it will be complicated, using sheafification many ...
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29 views

principal bundle and the associated bundle

Let $G\leq O(n)$ be a subgroup of orthogonal group. Let $\xi$ be a principal $G$-bundle. Let $\xi[\mathbb{R}^n]$ be the associated vector bundle. If $\xi$ is not a trivial bundle, can we obtain that ...
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33 views

A connection is uniquely determined by its action on global sections

Let $\mathcal{A}^i(E)$ be the sheaf of $C^{\infty}$ differential $i-$form with values in $E$, where $\pi: E \mapsto M$ is a complex vector bundle. We define connection on $E$ a $\mathbb{C}-$linear ...
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52 views

Coordinates at a singularity

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a smooth function. Let's assume that $f$ has a local minimum at $p \in \mathbb{R}^2$ and hence $| \nabla f|\ = 0$ at $p.$ Intuitively, one should be able to ...
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Eigenvalues of shape operators

Let A and A' be the two shape operators of the vector $\xi$, and if AX = $\alpha$ X, A'X = $\beta$ X, where $\alpha$ and $\beta$ are some functions. If after some calculation we obtain that X ...
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32 views

The period matrix of an $n-$dimensional torus

Let $\Omega$ be the matrix of the periods of the $n^{th}-$torus $T^n$ and $J$ an invertible alternating real $2n \times 2n$ matrix. Why $i\Omega^{*}J^{-1}\Omega$ is a Hermitian matrix?
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64 views

definition of derivative of a vector field

i try to define by myself the notion of differentiation of a vector field on a general manifold. I know that it is a classical subject and that there exist some answers as Lie derivative of a vector ...
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52 views

Orientation of a plane curve

Let $B(x_0)$ be an open unit ball in $\mathbb{R}^2$. Assume that $f:\overline{B(x_0)} \rightarrow \mathbb{R}^2$ is a diffeomorphism and $f(x_0)=y_0$. Then $f(\partial B(x_0))$ is a Jordan curve and ...
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Let $\gamma(t)=(\cos(3t),\sin(3t),4t)$ where $t\in \mathbb{R}$. Find $\{T,N,B\}$ and the torsion at the point $(-1,0,4\pi)$.

Let $\gamma(t)=(\cos(3t),\sin(3t),4t)$ where $t\in \mathbb{R}$. Find $\{T,N,B\}$ and the torsion at the point $(-1,0,4\pi)$. I'm not sure what $\{T,N,B\}$ is. I think $T=\gamma'(t)$, $N={1\over ...
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21 views

How is a regular surface in $\Bbb{R}^{3}$ different from an ordinary surface?

I have two books by Manfredo Do Carmo. He uses the concept of a regular surface in both books. In case you haven't read his definition, here it is: A subset $S \subset \Bbb{R}^{3}$ is a regular ...
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37 views

Linear independence of differential 1-forms

Let be $(E,\mathbb{K}),(F,\mathbb{K})$ Banach space and $U\subset E$ open set. If $f_1,...,f_n\in\Omega_1(U;F)$ differential 1-forms are linearly independents, where $$\Omega_1(U;F)=\{f:U\rightarrow ...