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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let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
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Completeness of a Riemannian manifold with boundary

I have some issues understanding the notion of completeness of a Riemannian manifold with boundary. In the case of Riemannian manifolds without boundary, I found that completeness is usually defined ...
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44 views

tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
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Any Straight Line Contained in a Surface is Asymptotic and Hyperbolic “Squares”

I have to prove that "any straight line $\alpha$ contained on a surface $S$ is an asymptotic curve and geodesic (modulo parametrization) of that surface $S$". Can I have hints at tackling this ...
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Are homeomorphic differentiable manifolds actually diffeomorphic?

Let $M$ and $N$ be two n-dimensional smooth manifolds.Suppose their underlying topological spaces are homeomorphic through $f$. Does $f$ automatically become a diffeomorphism with respect to the given ...
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Flat Surfaces in $\mathbb{R}^3$ Can Be Bent Only Along Straight Lines

This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, ...
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29 views

Christoffel symbols vanish in a system of normal coordinates.

I'm reviewing for a differential geometry exam and am getting stuck in a proof. This is based on question 4 from section 4-6 from little Do Carmo. Show that in a system of normal coordinates ...
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Structure of level sets of a noncritical point of a smooth function on a two dimensional domain

Let $\psi$ be a smooth function on a two dimensional simply connected domain $\Omega$ such that $\psi=0$ on the boundary $\partial \Omega$. Suppose $\rho$ is not a critical value of $\psi$ then it is ...
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1answer
22 views

2 dimensional Laplace's equation in polar coordinates

The problem asks you to get Laplace's equation in 2 dimensions in polar coordinates using the fact that $\operatorname{div}(\cdot)$ in two dimensional vector field could be written as $$\nabla \cdot u ...
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24 views

Finding relationship between Laplace-Beltrami operators of two spheres

Let $S$ and $T$ be spheres with radius $R_S$ and $R_T$ respectively. Define the diffeomorphism $\Phi:S \to T$ by $\Phi(s) = \frac{R_T}{R_S}s$. Given a function $u:T \to \mathbb{R}$, we can define ...
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Tangents of a Curve lie on a Cone

Prove that if all tangent vectors to the curve $α(t) = (3t, 3t^2, 2t^3)$ are drawn from the origin, then their endpoints are on the surface of a circular cone with the axis the line $x − z = y = 0$. ...
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Confusion about $\Delta_{S^n} u(x) = \Delta u\bigg(\frac{x}{|x|}\bigg)$, why do we need to divide by $|x|$?

Let $S$ be a sphere of radius 1. We know the formula $$\Delta_S u(x) = \Delta u\bigg(\frac{x}{|x|}\bigg)$$ holds for a function $u:S \to \mathbb{R}$ where $\Delta_S$ is the Laplace-Beltrami and the ...
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1answer
37 views

Link between a topological space and a manifold

A topological space is defined as a non-empty set $X$ together with a given collection of subsets $T$ (topology) of $X$, such that, (i) any union of these subsets is one of the subsets. (ii) any ...
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1answer
38 views

Del on Riemannian manifolds

I was supposed to figure out what $grad(div(e_r))$ is, where $e_r$ is the unit vector in spherical coordinates. In the following I assume $g:=diag(1,r^2,r^2sin^2(\theta))$ be the metric tensor. My ...
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A few identities on differential forms

I just started studying differentiable manifolds, and I've encountered a few allegedly simple questions that aren't really simple to me. If possible, I'd be very happy to see full proofs (without ...
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Proving a space is a manifold

Given a topological space defined as $A=A_1 \cup A_2$ with $A_1=\{(x,y) \in R^2 \space \space|\space \space x^2+y^2=1, x<0\}$, $A_2=\{(x,y) \in R^2 \space \space|\space \space |x|+|y|=1, ...
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How to prove a parallel $(u=u_0) $it self curvature?

My name is Gita, and I had aproblem with my math. and need help, I know that a parallel $C$ in a surface of revolution in $M$ be a geodesic if and only if $f'(u_0)=0$. and $C$ is non arc lenght ...
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81 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
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2answers
60 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
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33 views

Help Finding Orthonormal Frame and Coframe Based on First Fundamental Form

Given the metric (for $x^2<1$) $g=dx^2+2x dxdy+dy^2$ (in first fundamental form) I'm trying to find the orthonormal frame and its coframe I have found (I think) the orthonormal frame to be ...
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39 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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24 views

Prove that function F is diffeomorphism

I have a question in the book 'Elementary differential geometry' Prove that if a one-to-one and onto mapping $$F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$$ is regular, then it is diffeomorphism. I ...
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30 views

How to find a circle which approximates an unit speed curve at the origin?

A parametrization of a circle by arc length may be written as $$\gamma(t)=c+r\cos(t/r)e_1+r\sin(t/r)e_2.$$ Suppose $\beta$ is an unit speed curve such that its curvature $\kappa$ satisfies ...
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Reparametrize the curve problem in Differential geometry

Reparametrize the curve $\alpha(t)=(e^{t},e^{-t},\sqrt{2}t), \; \alpha: \mathbb{R} \rightarrow \mathbb{R}^{3}$, using $h(s)= \log(s)$ on $J:s>0$. Check the equation in Lemma in this case by ...
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Curve problem in Differential geometry

Find the coordinate functions of the curve $\beta = \alpha(h)$, where $x$ is the curve in $$\alpha (t)= \left( 1+\cos t, \, \sin t, \, 2\sin \frac{t}{2} \right)$$ for all $t$ and $h(s)=\cos^{-1}(s)$ ...
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1answer
26 views

need of smooth structure on manifold

I have seen that in the definition of a smooth function $f: M \to \mathbb{R}$, we firstly take $M$ to be a smooth manifold but i am not getting why do we need to take smooth manifold? The definition ...
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43 views

Lie derivatives and covariant derivatives (notation)

I am having troubling interpreting a particular expression in differential geometry. It arose in computing the Lie derivative along a unit normal, $n$, of the extrinsic curvature of a sub-manifold ...
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26 views

Level set of a real valued harmonic fucntion

Let $f$ be a real valued harmonic function defined on a neighborhood $U$ of origin in $\mathbb{R}^2$. And $f$ is such that its gradient vanishes at origin. Then how do i show that the set given by ...
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The magnetic monopole and the Hopf bundle

Consider the vector field $\textbf{B} = \frac{1}{\rho^2}\textbf{e}_\rho$ on $\mathbb{R}^3 - \{0\}$ where $(\rho, \theta, \phi)$ are the usual spherical coordinates and $\textbf{e}_\rho = ...
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Fundamental solution of the Laplacian on the surface of a cylinder

Does the Laplace operator have a fundamental solution on the surface of a cylinder in $\mathbb{R}^3$? Intuitively, I can visualize a function that diverges to $\infty$ at a point, decreases to a ...
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43 views

Klein Bottle Embedding on $\mathbb{R}^4$.

First of all, I am aware of the question in How to embed Klein Bottle into $R^4$ , which was inconclusive. Anyway, I've made some progress, but I still have a question. I am using Do Carmo's ...
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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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29 views

How to convert vector field from cartesian to spherical

I have a vector field $A ( r) = \omega \times r$, where $r=(x,y,z)^T$ and now I want to express this field in cylindrical coordinates. How do I do this?
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How to find $\nabla$ in spherical coordinates

I want to derive(!) just a few components like the $\hat{e}_r$ component of the divergence operator in spherical coordinates and the $\hat{e}_{\phi}$ component of the curl operator in spherical ...
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1answer
28 views

Derivative of Normal Vector Field

This is an example from Do Carmo (Example 4, page 139). Consider the saddle point $p=(0, 0, 0)$ of the hyperbolic paraboloid $z=y^2-x^2$ with parameterization $\mathbf x(u, v)=(u, v, v^2-u^2)$. It is ...
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42 views

Tangent at a singular point

I'm looking at this question If the tangent at the point $P$ with coordinates $(h, k)$ on the curve $y^2 = 2x^3$ is perpendicular to the line $4x = 3y$, find $(h, k).$ This is how I attempted it ...
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1answer
32 views

Why is $[\widetilde{v},\widetilde{w}]_p(f)=0$ when $f$ has a critical point at $p$?

Let $M$ be a smooth manifold and $f$ a smooth function $M\to\mathbb{R}$. Let $p$ be a critical point of $f$. We define the Hessian of $f$ at $p$ to be the symmetric bilinear functional $f_{**}$ on ...
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Ricci tensor of direct products of manifolds

Imagine I have a (Lorentzian) manifold with a metric $\left[ {\begin{array}{cc} g_{\mu\nu} &0\\ 0&g_{mn}\\ \end{array} } \right]$ Will the Ricci tensor be also block diagonal ...
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Diffeomorphism between a regular surface and the plane

Do Carmo states that (example 2, page 74) if $\mathbf x: U\subset\mathbb R^2\rightarrow S$ is a parameterization, then $\mathbf x^{-1}: \mathbf x(U)\rightarrow \mathbb R^2$ is differentiable. Why is ...
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Picture behind $SO(3)/SO(2)\simeq S^2$

Is there some kind of intuitive/waving hand argument to explain that $$SO(3)/SO(2) \simeq S^2 \; ?$$
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2answers
76 views

What does $d\bar z$ mean?

What does $d\bar z$ mean? For a manifold, given a local coordinate, $dx$ acts on tangent vectors and gives its corresponding components. What does $d\bar z$ do? The complex field is a one ...
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First and Second Fundamental Form Intuition

I was just wondering what various quantities relating to the first and second fundamental forms of a regular surface mean intuitively. First of all, another explanation as to what the first and second ...
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Differential geometry question on conformal mapping

Here is the Question http://postimg.org/image/gcte567fh/2f796778/ Math question so here is my issue I am getting an extra negative lurking around and so I get the following : $$||(k_1\cos^2\theta - ...
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Principle relative curvature of a surface

Find the principle relative curvatures of the surface (u^2 + v^2, u^2 - v^2, uv) at the point P : (u,v) = (1,1).? Find the relative curvature of the normal plane section at P which is tangent to the ...
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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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1answer
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Differential form on complex torus

Suppose $T= \mathbb{C}^n/\Gamma$ is a $n$-dim complex torus. How to prove that every exact $2$-form which has no $(0,2)$ component must be the image of a $(1,0)$ form? Is every torus Kahler? If the ...
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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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2answers
38 views

Lie algabra of R^n

Until now the only example of lie groups I have seen are subgroups of $GL_n$. Today I had the idea, that also $G=(\mathbb R^n,+)$ must be a lie group ($(\mathbb R^n,+)$ is a group with the ...
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Conditions for the partition of unity in general topology

While I am reading "An Introduction to Manifolds" by Loring W. Tu, I come to see the above theorem. I followed the proof but got a question on (ii). We are talking about smooth manifolds. Why do we ...
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Gaussian curvature and mean curvature sufficient to characterize a surface?

Is the knowledge of Gaussian and mean curvature (and thus of the principal curvatures) sufficient to characterize a surface uniquely? If not, is there another geometric quantity one can add to obtain ...