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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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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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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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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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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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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22 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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29 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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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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72 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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40 views

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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27 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 ...
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48 views

Proving the contracted Bianchi identity?

From Lee's book, the differential Bianchi identity states that for the Riemann curvature tensor, $$R_{ijkl;m} + R_{ijlm;k} + R_{ijmk;l}=0$$ The the proof is, contract on $i, l$, and then on $j, k$ ...
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Principle Relative Curvature

Calculate the principal relative curvatures of the surface y = xtan(z/a). I have checked in my textbook, but the only definition of relative curvature I could find is k = abs(d^2r/ds^2) and am unsure ...
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Contraction between basis vectors and basis one-forms

Discretion: The title may be misleading, because I am not certain whether the one-forms are actually basis one-forms. I always thought by definition, $dx^i (e_j) =\delta^i_j $. But, I am confused ...
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integration in five dimensions space part three

I am following this: integration in five dimensions space part two Maybe I need to simplify my question: Find the integration of $\int_{\partial S}-p_1dq_1\wedge dp_2\wedge dq_2$, where $S$ is the ...
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32 views

What is the Lagrangian tori?

I am looking for the definition of lagrangian tori for symplectic manifold $(M,\omega)$ ?
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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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Frenet formulas for curves in arbitrary Riemann manifold

As far as I understand, to have Frenet formulas one would need a curve, embedded in $\mathbb{R^n}$ and, desirably, naturally parametized. But there are homonomic notions of curvature and torsion of ...
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Slice of a coordinate system in a manifold

In the book - Foundations of differentiable manifolds and Lie groups by Frank Warner, the definition of a slice is as under. Suppose that $(U,\phi)$ is a coordinate system on $M$ (dimension $d$) with ...
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31 views

Geodesics and Curves on a Plane

Show that if a curve $C ⊂ S$ is both a line of curvature and a geodesic, then $C$ is a plane curve. Give an example of a line of curvature which is a plane curve and not a geodesic. (My thoughts: ...
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Various Parallels on a Torus

Consider the torus of revolution generated be rotating the circle $\{(x,y,z) \in \mathbb{R}^{3}: (x − a)^{2} + z^{2} = r^{2}, y = 0$ }, where $a > r > 0$, around the $z$-axis. The parallels ...
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The relation between principal curvature and curvature tensor?

To me, there are two systems of curvature of a surface, one is consist of 'principal curvature, mean curvature, Guass curvature, normal curvature' while the other is consist of 'curvature tensor'. I ...
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Application of Kodaira Embedding Theorem

I am going to give a talk on Kahler manifold. In particular, I will outline a proof of the Kadaira Embedding theorem. I also wish to give some applications of the theorem. One of the application ...
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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

integration in five dimensions space part two

I am following the discussion here: integration in five dimensions space I am doing this problem: Consider the differential form $$a=p_1 \, dq_1+p_2 \, dq_2-(p_1^2+p_2^2+q_1^2+q_2^2) \, dt\text{ in ...
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Formulas for Isometric Embedding / Minimal Immersion / Harmonic maps for a Helicoid in E3 / R3 in Higher Dimensional Euclidean Spaces.

Can anyone give me the actual formulas (or references to such formulas), in terms of the 2 parameters in E3 or R3, for a Helicoid to: Isometrically Embedding / Minimal Immersion / Harmonic Maps in ...
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Finding a formula for a $C^{\infty}$ 1-form $\omega$.

Let me elaborate more. Suppose that $(U, x^1, ... , x^n)$ and $(V, y^1, ... , y^n)$ are two charts on $M$ with a nonempty overlap $U \cap V$. Then a $C^{\infty}$ 1-form $\omega$ on $U \cap V$ has two ...
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36 views

Schwarzschild metric tensor normal vectors

The Euclidean Schwarzschild metric describing a manifold (a black hole, though this is not relevant to the question) is given by, $$\mathrm{d}s^2 = \left( 1-\frac{2GM}{r}\right)\mathrm{d}\tau^2 + ...
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k+1 Differential form

Consider the k-form given by, $ w = \sum_{i_{1}<i_{2}<...<i_{k}} w_{i_{1}i_{2}...i_{k}} dx^{i_{1}}\wedge dx^{i_{2}}\wedge...\wedge dx^{i_{k}}$ Define $k+1$ form $dw$ , the differential of ...
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The proof of cartan's magic formula

I want to use integral to prove the cartan's magic formula, i.e., it's enough to prove that for all small disk $D$ of dim=k=deg $\alpha$ in a manifold $M$, we have $$ \int_D L_X\alpha = \int_D ...
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Differentiable functions defined on a regular surface

First, recall a general definition of a differentiable function as follows. Suppose $f :D\subset \mathbb R^3 \rightarrow \mathbb R$. Then $f$ is differentiable at $\mathbf a \in D$ if there is a ...
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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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Umbilic Points of an Ellipsoid

I have an ellipsoid given by $S = \{ (x,y,z): \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$, for some fixed $a,b,c \in \mathbb{R}^{+} \}$. I need to find the umbilic points of ...
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18 views

Orthogonal Parametrization of a Regular Surface

I was just wondering whether or not it is always possible to parametrize a regular surface $S$ via a function $X$ of local coordinates $u$, $v$ such that $X$ is an orthogonal parametrization- that is ...
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The Lie derivative by an infinitesimal action is invertible at an isolated zero point

Let a compact Lie group $G$ act on a manifold $M$. Fix $X \in \mathfrak g$ and we write $X_M$ for the infinitesimal action of $X$. Assume that $p \in M$ is a zero point of $X_M$. Define a linear map ...
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Computing a Lie Bracket: General Questions

I'm asked to compute the following Lie Bracket: $\left [ -y \dfrac{\partial}{\partial x} + x\dfrac{\partial}{\partial y} , \dfrac{\partial}{\partial x} \right] $ on $\mathbb{R}^2$. Just writing it ...