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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Radius of Curvature

I was asked to show that the expression is constant in a circle : $\dfrac{\left[1+\left(\dfrac{\operatorname d \!y}{\operatorname d \!x}\right)^2\right]^{\frac 3 2}}{\dfrac{\operatorname d ...
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100 views

Conditions on a $1$-form in $\mathbb{R}^3$ for there to exist a function such that the form is closed.

What are the conditions on a $1$-form in $\mathbb{R}^3$ for there to exist a function such that the form is closed? More precisely, given a point, $p$, what are conditions on the coefficients of a ...
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70 views

Universal Property for Tangent Space

As far as I know there are basically three different approaches to the tangent space -all of them coming with advantages and disadvantages: Though the oldest one precisely represents the derivative ...
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1answer
47 views

Finding a space curve given some conditions on curvature and torsion

I have a space curve where the curvature is $\kappa$ and torsion $\tau = \kappa'$. An example of this would be a curve with curvature $\kappa = 1 - \cos s$ , $\tau = \sin s$. Is it possible to find ...
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1answer
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Tangent plane and normal

I need to find the tangent plane and a unit normal vector to $ r(u,v)=((2+\cos(u))\cos(v),(2+\cos(u))\sin(v),\sin(u))$ at $(2+\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}) $. $\frac{\partial r}{\partial ...
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Two curvature formulas when equal arc-length

all. So with a parametric curve $\vec{r}=\langle x(t),y(t)\rangle$, curvature is given by $$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$ When we have constant arc-length, an alternate ...
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38 views

parameterized ellipse, error in proof of a theorem?

A question from the book "Elementary Differential Geometry" from A Pressley Consider the ellipse $\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$, where $p>q>0$ The eccentricity of the ellipse is ...
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111 views

Find the maximum angle possible

$P$ is a point on the $Y-axis$ . Find the maximum possible value of $\angle APB$ where $A=(1,0)$ and $B=(3,0)$. Here is how I solved the problem. Suppose $P=(0,k)$ . Then using the cosine formula we ...
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56 views

Manifold locally looks like a open set but not as a euclidean space?

I am reading about manifolds from the book by Millman. He says that manifolds locally looks like a open set, but there is no canonical way to make M look like a Euclidean Space, and so we can't define ...
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1answer
51 views

parametrization the curve $x^2=4ay$

How can I able to parametrize the curve $x^2=4ay$ such that it becomes a ($i$) it becomes a regular curve. ($ii$)the parametrization becomes a unit speed parametrization. Actually I want to find the ...
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1answer
58 views

Parallel translation via $e$-connection

This question is concerned with Section 2.5. of Amari and Nagaoka's Information geometry book. Let me give some background first. Let $\mathcal{P}$ be the $n$-dimensional manifold of all (strictly ...
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196 views

Prove the curvature of a level set equals divergence of the normalized gradient

Suppose we have a function $\phi : \mathbb{R}^2 \to \mathbb{R}$, and a curve $\gamma:\mathbb{R}\to\mathbb{R}^2$ defined by a level set of $\phi$, ie. the codomain of $\gamma$ is ...
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53 views

laplace-beltrami and angular momentum operator

Im trying to understand this formula: In $\mathbb{R}^n$ we say that the laplacian can be seen as (radial part and angular part). : $$\triangle=(\frac{\partial^2}{\partial ...
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2answers
80 views

Divergence of a smooth vector field

I was studying about the divergence of a smooth vector field in the book "Calculus of variations and harmonic maps"by Urakawa. For a smooth vector field $X$, the divergence is defined by $div(X)(p) := ...
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1answer
67 views

Computing Normals from Metric Tensor

I asked a similar question on the Physics Stack Exchange, but unfortunately I have had no reply. It may be more suited for the Math section, as it focuses on the mathematical interpretation of GR. ...
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1answer
73 views

Unique manifold structure

I am reading the first chapter from the book - Foundations of Differentiable manifolds and Lie groups by Warner. There, he has given two statements to be proved as exercises. a) Let $M$ be a ...
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1answer
93 views

Defining a Hyperbolic Metric in a General Surface S

I hope someone can help me or give a ref. I'm trying to understand the general way in which one defines a hyperbolic metric on a given surface $\Sigma$ ( and, if not too complicated, on a manifold of ...
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48 views

Involutive distribution on a submanifold

Let $M$ be a manifold and $D:M\to TM:x\mapsto T_xM$ a distribution on $M$. According to Frobenius theorem, $M$ admits a foliation by maximal integral submanifolds of $D$ if and only if $D$ is ...
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Does there exist a vector field s.t. all orbits are dense on $\mathbb{R}^2$

Is there a complete vector field such that the all orbits are dense on a contractible manifold? For example, $\mathbb{R}^2$,the interior of a $n$-unit ball.
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1answer
36 views

What does it mean by saying that $u^n, J^n$ “$C^{\infty}$ converges” to u, J?

The question arose while reading the big book of McDuff & Salamon. Here $\Sigma$ is Riemann surface and M is compact symplectic manifold. Let $u^n(n\in \mathbb N), u : \Sigma \rightarrow M$ be ...
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1answer
76 views

How to proceed this computation with differential forms?

I've been studying Spivak's differential geometry book and he defines the exterior derivative of $\omega \in \Omega^k(M)$ in a coordinate system $(x,U)$ by $$d\omega = d\omega_{i_1\cdots i_k}\wedge ...
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1answer
34 views

How to show the pushforward is linear using equivalence classes of curves?

Let $M$ be a $C^k$ manifold of dimension $n$. I've constructed the tangent space at $a \in M$ as follows: first I've introduced the following equivalence relation in the set of maps $\gamma : ...
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27 views

How to prove that a fiber bundle restricted to a nilmanifold has a one dimensional fiber?

If you have an orthonormal frame bundle $\pi:P \longrightarrow M$ and $S \subset P$ is a nilpotent manifold, how do you prove that $\pi:S \longrightarrow M$ is a subbundle with fibre of dimension ...
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Why does the expression of a point on a manifold seem to assume a coordinate system?

This question came up when I was studying the definition of natural coordinate functions. In many books, such as O'Neil, natural coordinate functions are defined as $u_i(p) = p_i$ where ...
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2answers
67 views

Solving a differential equation $\displaystyle \frac{d \alpha}{dt}=w \times\alpha$

Let $\alpha$ be a regular curve in $\mathbb{R}^3$ such that $\displaystyle \frac{d \alpha}{dt}=w \times\alpha$ for $w$ a constant vector. How can we determine $\alpha$ ? $\displaystyle w ...
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4answers
230 views

Tangent plane passes through origin

This is from a section in my course book on elementary differential geometry: Since the tangent plane $T_p S$ of a surface $S$ at a point $p \in S$ passes through the origin of $\mathbb{R}^3$, it ...
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1answer
54 views

Image of an Immersion

I cannot prove any conclusion of this problem. Can anyone please help me? Let $f:M\to N$ be an immersion of $M$ into $N$ and dim $M=\dim N$.Prove or disprove that $f(M)$ is a submanifold. Thanks for ...
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1answer
137 views

Show isometry of flow on a compact Riemannian manifold where the vector field is Killing

Let $(M,g)$ be a Riemannian manifold, $\nabla$ the Levi-Civita connection of $g$. A vector filed $V$ on $M$ is called a Killing field if for every $p\in M$ and every $X,Y\in T_p M$, $$ g(\nabla_X V, ...
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1answer
56 views

Notation for coordinate-free Tautological Form definition

Reading Ana Cannas da Silva's book, I found the following step defining the tautological form (the "$p_i\wedge dq^i$" form) in a coordinate-free manner. Let $X$ be a given manifold, its cotangent ...
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32 views

A question about complex polarization

Let $M$ be a smooth manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar P \cap ...
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182 views

Geodesics on torus

Describe the geodesics on Torus $$\sigma (u,v)= ((a+b \cos u)\cos v, (a+b\cos u)\sin v, b\sin u)$$ First fundamental form for torus is $$b^2 du^2 +(a+b \cos u)^2dv^2$$ Consider unit-speed ...
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2answers
67 views

Geodesics on spheroid

Describe the geodesics A Spheroid obtained by rotating the ellipse $\frac{x^2}{p^2}+\frac{z^2}{q^2}=1$ around the z-axis where $p, q\gt 0$ Please explain this question explicitly. Thank you:)
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1answer
53 views

Integration of forms over a smooth orientable manifold

Why does one needs to assume that a $n$-form has a compact support to be able to define its integration over a orientable $n$-manifold? I'm referring to the following text: ...
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39 views

How to compute saddle point index using sourcing flow lines?

Prove $Index_{p}(\bigtriangledown f)$= "dimension of sourcing flow lines from p" ,where p is a critical point. Attempt Near $ p \in Cr(f) $ in some coord. $ f(x) - f(p) = $ $\sum_j x_j^2 - \sum_k ...
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2answers
81 views

Integral of a differential 1-form along a curve (clarification on the definition)

Let's denote with $(e_1,\dots,e_d)$ the usual basis of $\Bbb R^d$, and with $({e_1}^*,\dots,{e_d}^*)$ the dual basis of its dual space $\Bbb {(R^d)}^*$. Let $U$ be an open subset of $\Bbb R^d$ and ...
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0answers
47 views

Canonical “orientification” of a manifold? Canonical complexification of a manifold?

Maybe this is a silly question(i'm pretty new to both geometry and category theory) but i was wondering: 1)Consider the category of orientable smooth manifold on $\mathbb{R}$, if you forget the ...
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1answer
41 views

Embedded and Non-Parametric Surface definition

What does it mean for a minimal surface to be embedded? For example the Scherk surfaces? How would I define what 'an embedded surface' is? And also what does it mean for a surface to be ...
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25 views

Derive locally conformal neighborhoods between a Riemann surface and a diffeomorphic image of it

Consider a regular Riemann surface $M \subset \mathbb{R}^3$. A deep theorem in differential geometry states that any two regular surfaces are locally conformal [1]. The proof of this theorem consists ...
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83 views

Existence and uniqueness on this semi-linear parabolic PDE

I want to know whether the existence and uniqueness of a classical solution can be found about this question: Find a classical solution $u : [0,T]\times [0,\infty] \rightarrow {\mathbb R}$, such ...
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votes
1answer
76 views

Constant Speed of Geodesics

Let V be the set of smooth functions $ f : [0,1] \rightarrow \Bbb R $ such that $ \int_0^1 f(t) dt = k $. If $ F : V \rightarrow \Bbb R $ is given by $ F(f) = \int_0^1 f(t)^2 dt $, then show that the ...
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2answers
144 views

Geodesic equations and christoffel symbols

I want to learn explicitly proof of the proposition 9.2.3. Which books or lecture notes I can find? Please give me a suggestion. Thank you:)
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39 views

Central subgroups …

Have a question about the central subgroup. E is an elation group. Let $E_{x,y}$ be a stabilizer of points $x$ and $y$. Then the group $E_{x,y}$ is a central subgroup both $E_{x}$ and $E_{y}$, ...
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1answer
47 views

Euler-Lagrange Eqn to find eqn of a straight line

I'm trying to see how we use the E-L equation \begin{equation} L_x(t,q(t),q'(t))-\dfrac{d}{dt}L_v(t,q(t),q'(t))=0 \end{equation} to find the shortest distance between two points in the Euclidean ...
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1answer
43 views

Wave equation on a compact Riemannian surface without boundary: no mass conservation?

Consider a compact, smooth Riemmanian surface $\mathcal{S} \subset \mathbb{R}^3$ without boundary. I would like to solve the wave equation: $$u_{tt} + \Delta_{\mathcal{S}} u = 0$$ under the ...
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1answer
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Is it possible to “lower” a $\pi_1$-invariant differential function defined over the universal covering manifold to the base one?

Consider a differentiable manifold $M$ and its smooth universal covering $\pi:\tilde{M}\rightarrow M$. There is a canonical action of the fundamental group $\pi_1(M)$ on the covering manifold ...
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1answer
41 views

How to understand the second fundamental form is Tensorial?

From wikipedia. The second fundamental form is defined by $$II(u,\ v)=\left \langle \nabla_u v,\ n \right \rangle$$ where $\nabla$ is the Levi-Civita connection of the ambient manifold $M$, and $u, ...
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31 views

On implicit function theorem

We consider a manifold $M$ given by $f_{j}(x,y,v,w)=0$, $j=1,2$, where $f_{1,2}$ are smooth functions on $R^{4}$ verifying $rank \frac{D(f_{1},f_{2})}{D(x,y,v,w)}=2$ in every poin of $M$. If ...
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104 views

Best closed convex surface fitting N points in 3D

First. It's easier to understand the problem by describing the application where it arises from. We have a convex body $B$ in $\mathbb{R}^{3}$ and measure points on its surface. The measurements are ...
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63 views

tubular neighborhoods - problem

Consider $\alpha:[0,L]\to \mathbb R^3$ a smooth regular simple closed curve, arc-length parametrized. Denote its trace by $\gamma$. Let $\epsilon >0$ be a real number and $N_\epsilon (\alpha)$ the ...
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1answer
111 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 ...