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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Explicitly reconstruct conformal mapping from conformal factor

Consider two smooth, compact surfaces $\mathcal{S_i} \subset \mathbb{R}^3$, with Riemannian metrics $g_i$, $i=1,2$ and a conformal mapping $f:S_1\rightarrow S_2$. Suppose we know the conformal ...
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25 views

Can the image of chains on a smooth manifold be thought of as a Borel $\sigma$-algebra?

Volume forms on smooth manifolds have a nice interpretation as measures, but what takes the place of the Borel $\sigma$-algebra? In particular, if we let $\mathcal{M}$ be a smooth manifold and ...
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1answer
51 views

Why is this function not injective?

I'm working through some of Do Carmo's Differential Geometry excercises. In one of those excercises he asks if the map $f(u,v)= (u+v,u+v,uv)$ with $u>v$ is a parametrization for the plane. In the ...
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39 views

Dual space (Wikipedia)

I am struggling to understand something on Wikipedia: ''If $V$ consists of the space of geometrical vectors (arrows) in the plane, then the level curves of an element of $V^*$ form a family of ...
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38 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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67 views

Is this counter-intuitive result actually correct?

I was trying to calculate the surface area of part of a sphere. The result seems counter-intuitive. I have found that the surface area of a sphere increases linearly. Consider the circle, centre ...
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38 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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69 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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1answer
34 views

Orthogonal connection on tangent bundle

What does orthogonality of connection mean in coordinate way? As I understand, a connection $\nabla: \Lambda^1M \rightarrow \Lambda^1M \otimes \Lambda^1M$ is torsion-free iff in any local coordinates ...
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41 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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21 views

Planes and surfaces and normal vectors?

Is a plane the same thing as a surface? and is the normal vector the same at every point on both??
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1answer
37 views

The principal curvatures of a surface of revolution

The principal curvatures of the surface at a point is defined as the maximal and the minimal curvature among all normal sections. It's claimed (say, on Stillwell's Geometry of Surfaces) that for a ...
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115 views

Analogue of a parabola on a sphere?

Parabola: the set of points in the plane that are equidistant from a line, called the directrix, and a point, called the focal point, not on the line. Suppose we try to replicate this on a sphere: ...
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40 views

Can Vanishing Cycles be Described as Fibers over Critical Points in a Lefschetz Fibrations?

I'm trying to see if it makes sense to see vanishing cycles in a Lefschetz fibration as the fibers over critical points. A Lefschetz fibration $f: M^4 \rightarrow X$ , where $M^4$ is a smooth ...
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1answer
24 views

locate critical points/values and show where function is a regular surface

Let $f(x,y,z) = (x + y + z - 1)^2$. a. Locate the critical points and critical values of $f$. b. For what values of $c$ is the set $f(x,y,z) = c$ a regular surface? a. So, to locate critical points ...
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43 views

Solving a certain differential equation when assuming a surface of revolution is minimal

The problem is the following: Consider the surface of revolution $$ \textbf{q} (t, \mu) = (r(t)\cos(\mu),r(t)\sin(\mu),t) $$ If $\textbf{q}$ is minimal, then $r(t) = a\cosh(t)+b\sinh(t)$ for $a,b$ ...
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1answer
48 views

Recovering a frame field from its connection forms

I have a faced a research problem where I would need to recover a frame field given its connection forms. More precisely, I begin with an orthonormal frame field (given by data) in $\Re^3$ written as ...
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1answer
34 views

Winding number of a linear transformation?

I know that I am computing something incorrectly. I am trying to compute the index of a positive determinant linear bijection. The form I am using is $\omega = \frac{-y dx + x dy}{x^2 + y^2}$. I ...
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1answer
25 views

Why use Gauss and mean curvature to characterize a surface's deviation from being “flat” at one point?

We know for a 2-dimensional surface there are two orthogonal principal directions at every point, where the principal curvatures $\kappa_1$ and $\kappa_2$ are the two ends of the curvature spectrum ...
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2answers
50 views

A question on tangent vector fields

I'm currently taking an introductory course in Differential geometry of curves and surfaces. I have a question on vector field, $\textbf{w}$: Given $p \in S, ...
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1answer
34 views

Isometry of spheres/hypersurfaces and more generally Riemannian manifolds.

Let $M$ and $N$ be two spheres (of different radius) in $\mathbb{R}^n$ of dimension $n-1$. Suppose there is a Riemannian isometry between them (so a diffeomorphism and isometry). Then distances must ...
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28 views

Can we define a induced metric like this?

Let $\Sigma_r$ be a topological sphere in a 3-dimensional asymptotically flat Riemannian manifold $M$ with metric $g$, $\{\frac{\partial}{\partial x^i}\}, 1\leq i\leq3$ is the standard coordinate ...
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40 views

Tensor differential equations

I am reading Ringstrom's book The Cauchy problem in General Relativity, But I don't really understand Chapter 12 associating to tensor equations. I want to read some other material about this. Could ...
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1answer
27 views

Second fundamental form without orientability?

Let $F$ be a $C^2$-hypersurface (or $n$-manifold) embedded in $\mathbb{R}^{n+1}$. Suppose $F$ is not orientable. Since I cannot choose a continuous global normal field, what consequences does this ...
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4answers
87 views

Is there a shorter path to these results?

I'm a student of Physics, however I usually study mathematics on texts aimed at mathematicians to gain a deeper understanding. Currently I'm studying differential geometry on Spivak's book and one of ...
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1answer
41 views

About “good” covers of manifolds

Let $M$ be compact manifold of dimension $n$. I am wondering if a certain type of covering of M by coordinate charts exists (this is not the usual good covering theorem but seems related to it). I ...
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1answer
39 views

Is a surjective submersion map proper?

Let $M$ and $N$ be two smooth manifolds and $f$ a surjective smooth map from $M$ to $N$ which is a submersion. If for any $p \in N$, $f^{-1}(p)$ is compact, then is $f$ necessarily proper, that is, ...
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1answer
87 views

The Curvature Tensor

I present three different ways I've seen the Riemann curvature written: $R(X,Y)Z=D_XD_YZ-D_YD_XZ-D_{[X,Y]}Z$ $R(e_c,e_d)e_b=D_{e_c}D_{e_d}e_b-D_{e_d}D_{e_c}e_b-D_{[e_c,e_d]}e_b$. $R^{\rho}_{\space ...
3
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1answer
99 views

Showing to be Unit-sphere

First and second fundamental forms are both $du^2 +\cos^2 u dv^2$ I want to show that the surface is a part of the unit sphere. What I did is following; $E=L=1$ $F=M=0$ $N=G=\cos^2 u$ ...
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47 views

Are there any results linking hyperbolic operators and Pseudo-Riemaniann geometry?

In the case of a $n$-dimensional pseudo-riemannian manifold one has a symmetric bilinear form, $h$, with signature $(1,n)$ and the analogous operator to the Laplace -Beltrami operator,$\Delta$, is the ...
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41 views

Why the Oloid is developable surface?How to prove it?

It is well known that the Oloid is developable surface,but why?How to prove it?
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26 views

Components of a Vector Field

This may be quite a petty question, but it's been bothering me for a while. So for the vector fields $X$ and $Y$, we can write them in component form as $X=X^{\mu}\frac{\partial}{\partial ...
3
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1answer
39 views

How to understand structure groups?

I'm studying fiber bundles and I'm somewhat confused on how Structure Groups appears. The definition of fiber bundle I have is the following: A bundle is a tuple $(E,B,\pi)$ where $E,B$ are ...
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20 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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30 views

Gradient flows of functionals on manifolds

I'm reading some literature of ricci flows and on my way through it I quickly stumbled upon the gradien flow - one of the geometric flows. After searching on other books I found rigor definition of ...
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24 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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149 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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2answers
110 views

Expression for Laplace-Beltrami on sphere?

Is there a good expression for the Laplace-Beltrami on a function $u$ on a sphere or a circle of radius $R>0$ in terms of the Laplacian on ambient space? There is a formula on Wikipedia for the ...
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13 views

Finding the Bessel function from its derivative

I have a situation: $A_k\frac{\partial J_m(k\rho)}{\partial \rho}=0$. where $k=k_1$ for $0\leq\rho\leq a$ and $k=k_2$ for $a \leq \rho \leq \Lambda-a$ with $a,\Lambda\leq \infty$. Can I proceed with ...
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32 views

Example of 1-dimensional hypersurface in $\mathbb{R}^2$ which is compact?

Is there an explicit example of a $1$-dimensional $C^k$ hypersurface in $\mathbb{R}^2$ which has no boundary and is compact? I know of a circle, but want something like an interval.
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1answer
59 views

History of the Enneper Surface

I was just wondering whether anyone could tell me more about the Enneper surface and its history (why it is important historically in the development of mathematics), or where to go in order to learn ...
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1answer
36 views

Unitary connection and Hermitian connection

I am confused with the two notions. I basically understand Hermitian connection: it is a complex analog of metric-compatible connection on $TM$, a connection that preserves the hermitian metric $h$ ...
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32 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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1answer
37 views

Smooth Embedding

I need help showing that the following smooth map is not a smooth embedding: $f:\mathbb{S}^1 \to \mathbb{R}$ defined by $f(z)= \operatorname{Re}(z)$. I know that this map is not a submersion because ...
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60 views

If there exists a diffeomorphism between two surfaces, what is the relation between Laplace-Beltrami operators on the surfaces?

Let $S(0)$ and $S(t)$ be a hypersurface in $\mathbb{R}^n$. Suppose there is a diffeomorphism $F^0_t:S(0) \to S(t)$. Suppose we have the Laplace-Beltrami operator $\Delta_{S(\cdot)}$. Let $u:S(t) \to ...
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31 views

What's the first fundamental form of a regular surface in complex coordinates and how to get it?

Precisely, the first fundamental form of a regular surface is given by $$ds^2=Edx^2+2Fdx\ dy+Gdy^2.$$ What's the form of $ds^2$ in complex coordinates $z=x+iy$.
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Compatible connection on the associated vector bundle

Assume $E\rightarrow X$ is a holomorphic vector bundle of rank $n$ with a linear connection $\nabla=\nabla^{1, 0}+\bar\partial_E$ which is compatible with the complex (somewhere the literature says ...
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1answer
14 views

If $S$ is a $C^k$ hypersurface, is $S\times (0,\infty)$ a $C^k$ hypersurface too?

Let $S$ be an $n$ dimensional $C^k$ hypersurface in $\mathbb{R}^{n+1}.$ Is $S \times (0,\infty)$ also a $C^k$ hypersurface (in $\mathbb{R}^{n+2}$)? I don't know what the chart map should be...
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41 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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25 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 ...