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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288 views

Boundary of unit square is not a smooth submanifold of $\mathbb{R}^2$?

I've read some of the answers to related questions to this, but this is an idea I've been grappling with for a while and still can't fully get my head around. $\mathbb{S}^1$ is a smooth manifold, and ...
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54 views

Differential Geometry: Angle between coordinate lines

Given the surface: $$x= u(3v^2 - u^2 - 1/3), y= v(3u^2 - v^2 - 1/3), z= 2uv\ .$$ Find the angle between the coordinate lines. I'm not entirely sure what is meant by coordinate lines or how to find ...
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1answer
31 views

Differential Geometry Angle/First Quadratic Problem

Find the angle between the curves $v = 2u + 1, v = -2u +1$ on a surface with the first quadratic form: $E = 2, F = 1, G = 4$. I know I should probably use the $cos(\theta)=\frac{T_1(0)\cdot ...
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1answer
61 views

Geometric interpretation of $\partial/\partial z$

My understanding is that analytic derivative ,$\partial\phi/\partial z$, and anti-analytic derivative ,$\partial\phi/\partial\bar{z}$, are resp. tangent and normal to the curve $\phi$. Am i right?can ...
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1answer
72 views

An immersive map is locally left invertible

Question: Suppose that $m < n$, that $U$ is an open set in $\mathbb R^m$ and that $f : U \rightarrow \mathbb R^n$ is a $C^1$ function that has rank $m$ everywhere in $U$. Show that for every ...
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1answer
223 views

(Determinant of) Hessian in local coordinates

Let $f\colon M\to \mathbb{R}$ be a smooth function on a manifold $M$ with a critical point $p$. We define its Hessian at $p$ via $H(u, v)=(UVf)(p)$ where $u, v\in T_pM$ and $U$ and $V$ are vector ...
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141 views

Given Constant Ratio of Torsion to Curvature, Show Tangent times Constant Vector is Constant

Let $r(t)$ be a unit speed curve such that for all $t$, $\frac{\tau(t)}{\kappa(t)}=\cot(\theta)$ for some $0 < \theta < \pi$. Show that there is a constant vector $a$ satisfying $T(t) * a = ...
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45 views

PDEs along parametrized curves in $\mathbb R^2$

For Riemannian surfaces in $\mathbb R^n,n\geq 3$, the Laplace-Beltrami operator can be expressed in terms of the Riemannian metric and one can consider evolution equations (e.g. heat diffusion, wave ...
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1answer
22 views

Diferentiating between interna and external Angle in a cycle directed edge list

I've got a cycle list of directed edges that delimite an interior and exterior and of course I can get the small angle between any consecutive edges with the dot product definition with $cos(\alpha) = ...
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1answer
447 views

Show the cylinder is a regular surface

Show that the cylinder $(x,y,z) \in R^3; x^2+y^2=1 $is a regular surface and find parameterizations whose coordinate neighborhoods cover it. I'm going to be honest I saw this answer but I don't quite ...
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1answer
83 views

Curvature of saddle by definition

I'm trying to compute the principle curvatures of the saddle $M$ defined by $z= y^2 -x^2$ at the point $p = (0,0,0)$, but I know my computations are wrong. Maybe you can help to see where I went ...
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2answers
174 views

Given Unit Speed Curve on a Sphere, Show the Curve has Constant Curvature

Let $r(t)$ be a unit speed curve on a sphere $x^2+y^2+z^2=R^2$. Show that the curve $c(t)=\int^t_0 r(u) du$ has a constant curvature $\frac{1}{R^2}$ I am still a little shaky with this stuff, so I ...
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1answer
142 views

Projection map between the Stiefel manifold and the Grassmanian

I am trying to show that the projection map $\pi: V_{k}(\mathbb{R}^{n+k}) \rightarrow \mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ is a fiber bundle with fiber $O(k)$, the group of orthogonal $k \times ...
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1answer
105 views

Is this a geodesic?

Let $(M,g)$ be a riemannian manifold. Let $p$ in $M$ and $v,v_{0}$ two vectors in $\mathrm{T}_{p}M$. I am looking at the curve $$ \gamma \, : \, t \, \longmapsto \, \mathrm{Exp}_{p}(tv+v_{0}) $$ ...
4
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1answer
98 views

Calculus on Manifolds - operational point of view

I'm a student of Physics and I've been studying manifolds and calculus on such objects for a time. Usually when we deal with vector calculus there are books that bring one operational point of view. ...
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2answers
46 views

Distorted Unitary matrices

Let $U$ be an unitary and $D$ be a diagonal matrix. We know that for all vectors $v$ on the sphere $Uv$ is on the sphere and, $$\langle Uv,Uv\rangle=\langle v,v\rangle.$$ What are the vectors $v$ on ...
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124 views

What is the generalization of Gauss's Theorem to a manifold?

In a (pseudo-)Riemannian manifold with constant basis vectors, one certainly has that the integral of the divergence of a tensor field $T$ over a submanifold $\Omega$ is equal to the integral over the ...
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2answers
120 views

Meridians of surfaces of revolutions

First off, I know there is another question asking the same thing, but that one was concerning where to start, whereas for this one, I'm almost complete, but I can't get something at the end to ...
3
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2answers
629 views

parametrization of the hyperboloid of two sheets

Find the parametrization for the hyperboloid of two sheets${(x,y,z) \in \mathbb{R}^3}; -x^2-y^2+z^2=1$. Ok so I saw two answers for this question: $x(u,v)=(\sinh u \cos v, \sinh u \sin v, \cosh u)$ ...
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46 views

How do I convert OS coordinates (X and Y) to longitude and latitude coordinates?

How do I convert OS coordinates (X and Y) - Eastings and Northings to longitude and latitude coordinates? For example X and Y below X (Eastings): 347904 Y (Northings): 287484
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1answer
47 views

Differential equations of a plate with spherical mass load?

I would like to know the equations describing a plate surface being curved and stressed by a mass (you know, like a ball on a stretched sheet). I'm just curious :). I feel a bit confused about tensor ...
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2answers
258 views

Prerequisites for Freedman's proof of the 4-dimensional Poincaré conjecture

I have a good understanding of differential geometry, enough at least to understand many details of Hamilton & Perelman's approach to the 3-dimensional Poincaré conjecture. I have no such ...
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1answer
44 views

Show that $[(x,y,z)] = \dfrac{xy + xz + yz}{x^2 + y^2 + z^2}$ is smooth.

This is a function from the real projective space $f: RP^2 \to R$. I need show $f \in C^{\infty}$. Again, $[(x,y,z)] = \dfrac{xy+xz+yz}{x^2 +y^2 + z^2}$ I think it's pretty simple, but I need ...
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58 views

Immersions when the target space isn't a differentiable manifold (but *almost* is)

I've come across this situation in a number of places but it's most glaring in the lecture notes I'm currently reading. (PCMI lectures on the geometry of outer space). We have a map from a circle ...
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1answer
35 views

surface curvature

I would like to proof the existence or the non-existence of a finite surface which has 2 different radius of curvature $R_1$ and $R_2$ that are: constant on the whole surface finite different each ...
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1answer
59 views

Compact Implies Closed

In general, i believe, it is not true that Compact Implies Closed. At least it is true that Compact does not imply closed and bounded. However, in case of differentiable manifolds, since they are ...
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1answer
78 views

Extending pullback of a vector field

Consider the vector field $\frac{\partial}{\partial x_1}$ on $\mathbb{R^2}$. Let $\psi_N : S^2 \setminus\{N\} \to \mathbb{R^2} $ and $\psi_S : S^2 \setminus\{S\} \to \mathbb{R^2} $ be the ...
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1answer
122 views

Homework: calculation about differential form

Here is the question: Let $\omega = A dy\wedge dz + B dz \wedge dx + C dx \wedge dy$ in $\mathbf{R}^3$, and $d\omega = 0$. Denote \begin{eqnarray} \alpha = \int_0^1 tA(tx,ty,tz)dt\cdot(ydz-zdy)\\ ...
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1answer
76 views

Calculate the integral of a 2 form

I am trying to compute the integral $$ \int\int_{S}\frac{1}{x}dy\wedge dz+\frac{1}{y}dz\wedge dx+\frac{1}{z}dx\wedge dy $$ over an ellipsoid given by $$ ...
4
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1answer
148 views

Isometry vs. measure preserving?

Consider functions between two measured metric spaces. What is the relation between an isometry and a function which preserves the measure of subsets? This question arose in my head as I thought ...
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2answers
97 views

Using a non-zero wedge product to write a set of vectors as a linear combination of another set of vectors in a finite dimensional space.

Question: Let $V$ be a finite dimensional vector space, and let $ \{ v_1, ..., v_r\}$ and $\{w_1, ..., w_r\}$ be two sets of vectors in $V$. Suppose that $\sum_{i=1}^{r} v_i \wedge w_i = 0$, and ...
2
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2answers
92 views

Proving a submanifold of $SL_2(\mathbb{R})$

I already showed that $SL_2(\mathbb{R})$ is a 3-dimensional manifold. Now I want to show that the subspace $E$ of symmetric matrices whose eigenvalues are positive in $ SL_2( \mathbb{R})$ is a ...
2
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2answers
280 views

Grassmannian, Plucker coordinates

In which books can I find something about the grassmannian and the plucker coordinates ?
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1answer
95 views

How to obtain a diffeomorphism between $\mathrm{SL}(2,\mathbb{R})$ and $ (\mathbb{C}\!\smallsetminus\!\{0\})\times\mathbb R$?

Could someone please give me a tip on how to show that the map $\mathrm{SL}(2,\mathbb{R}) \to (\mathbb{C}\setminus\{0\})\times\mathbb{R}$ \begin{equation}\begin{pmatrix} a & b\\ c & d ...
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0answers
30 views

I need help with the derivation of the equation of a tangent line at a point on a curve, and the arc length parameter $s$.

I'm having difficulty seeing how under the arc-length parameterization the equation of the tangent line $$\frac{X-x}{dx}=\frac{Y-y}{dy}=\frac{ Z-z}{dz}=u$$ can be written as ...
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0answers
98 views

Surface of revolution

This problem is from Dubrovin's Modern Geometry (Problem 8, Exercise 8.4). Show that the only surfaces of revolution with zero mean curvature are the plane and the catenoid (which is the surface ...
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5answers
103 views

Surface all of whose normals intersect at a point

I am new to differential geometry and encountered difficulty when trying to solve the following problem from Dubrovin's Modern Geometry It's the first problem in exercise 8.4: Find the surface ...
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1answer
85 views

Morse height function for general compact manifold

Can you give me the form of the height function for any compact manifold embedded in the reals? Maybe the projection of the parametrization onto a basis vector ex. For the n-sphere is ...
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1answer
146 views

(Morse) non-degenerate iff transverse to the zero section

So Morse $f:M\to \mathbb{R}$ has nondegenerate critical point p iff $df|_{p}\pitchfork 0$-section. Attempt nondegenarate p iff Hessian has full rank at p iff ...
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2answers
61 views

showing a surface is smooth and computing first fundamental form

let $U \subset R^2$ be a non-empty open set and let $f:U \to R$ be a smooth function. If $Y \subset R^3$ given by $Y={ (u,f(u,v),u+v) \in R^3|(u,v) \in U}$ show $Y$ is a smooth surface and compute ...
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29 views

Help modeling 3d vector field

I'd like some help in finding the correct mathematical description of the stuff below: A sheet is deformed by a mass on it (like in one of those pictures showing the effects of General Relativity). ...
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1answer
53 views

Isometric map of geodesic

Assume a Riemann manifold $(M,g)$ and a smooth map $\sigma:M\times M\rightarrow M$, $(m_{1},m_{2})\rightarrow \sigma_{m_{1}}(m_{2})$, such that: $\forall m\in M$ $\sigma_{m}:M\rightarrow M$ is an ...
2
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1answer
271 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
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4answers
314 views

Topics in Differential geometry $\cap$ Algebraic geometry

I find (both: differential and algebraic) geometry fascinating. I'm just beginning my graduate studies, but I'd like to know some topics/theorems in the intersection of these two (since I don't really ...
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200 views

Who invented the Riemann Sphere?

I have seen suggested that someone other than Riemann first came up with the Riemann Sphere. Is this correct? And if so, who did invent it?
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38 views

Is the boundary of $ Q=\bigcup_{t \in (0,T)}\{t\}\times S(t) $ empty when $S(t)$ is a boundaryless hypersurface?

For each $t$, let $S(t)$ be a compact hypersurface in $\mathbb{R}^n$ with $\partial S(t) = \emptyset$. Consider $$ Q=\bigcup_{t \in (0,T)}\{t\}\times S(t) $$ Is the boundary of $Q$ empty? I don't ...
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1answer
86 views

Injective immersion (between smooth manifolds) that is no homeomorphism onto its image

Is there an injective immersion between smooth manifolds that is no homeomorphism onto its image? With smooth I mean $C^\infty$-manifolds and of course also the immersion should be $C^\infty$.
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180 views

Cartesian vector field to vector field

Ok so I have a given vector field in Cartesian coordinates, say \begin{align*} \textbf{v}(x,y)=\frac{dx}{dt}\hat{\textbf{e}}_{1}+\frac{dy}{dt}\hat{\textbf{e}}_{2} \end{align*} Where $dx/dt$ and ...
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2answers
290 views

When does a SES of vector bundles split?

Given a short exact sequence of smooth vector bundles, $$0\to A \to B \to C \to 0$$ on a manifold $M$, it is an easy exercise that sequence splits. One approach is to pick a Riemannian metric on ...
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306 views

Chain rule with covariant derivative

Let $\mathcal{M}$ be a $n$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. Consider the following function: $$\tilde{F}(v) = \operatorname{d exp}^{-1}_{p} ...