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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Vector Laplace Beltrami operator on surface tangent and surface normal vector field

Consider a closed, compact, embedded surface $f:M \rightarrow \mathbb{R}^3$ and a vectorfield $X$ on the surface that can be decomposed in the surface frame basis $\{e_1,e_2,e_3\}$, where ...
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Integration by substitution and separation of variables.

Let's say I want to integrate over a sphere $S^2$. Take $f \in (L^1(S^2),dS)$, then we have that $$\int_{S^2} |f| dS = \int_{S^2} |f| \sin^2 (\theta) d \theta d \phi < \infty,$$ right? Now, ...
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How to embed Klein Bottle into $R^4$

I am using Do Carmo's Riemannian Geometry, and struggling to solve a problem. The problem is: Show that the mapping $G:\mathbb{R}^2\to\mathbb{R}^4$ given by $$G(x,y)=((r\cos y+a)\cos x,(r\cos ...
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Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
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Canonical connection on $CP^n$

I have heard something along the lines of "There is a canonical $U(1)$ connection on $CP^n$" and I am trying to understand what that means. First I suppose that the sentence refers to a line bundle ...
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A construction on principal bundles

In a paper the principal $Sp(1)$-bundle $P$ over $S^4$ is introduced as follows: let $Sp(1)\times Sp(1)\hookrightarrow Sp(2) \xrightarrow{\pi} S^4$ be the spin structure on $S^4 $. The principal ...
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Applications of Geometry to Computer Graphics [on hold]

How is differential geometry (or any type of theoretical math) related to computer graphics and/or computer programming? A friend of a friend of mine has only a bachelors degree in pure math and got ...
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2answers
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The meaning of $\omega$ = df?

I'm reading Algebraic Topology by William Fulton and I am a little lost about 1-forms, which I assume to be the number of differentials according to the number of dimensions (correct me if I'm wrong. ...
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1answer
262 views

Differential forms and a chain rule

Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$. Let $Q\in U$ ...
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Derivative of terminal state w.r.t. the inital conditions.

Let $x\in R^n$ and consider the system $$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$ and suppose that we know it's exact or very accurate solution $x(t)$ for the time interval $[0,T]$. I'm ...
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How to know if a tangent bundle is trivial from its defining equations

In this question, I am considering only regular manifolds. A Trivial Bundle The circle $S^1$ is known to have a trivial tangent bundle. As a subset of $\mathbb{R}^4$, the tangent bundle of $S^1$ ...
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1answer
79 views

Caracterization of isometries that preserve time-orientation in $\Bbb L^3$

First of all, I'm considering $\Bbb L^3$ with the convention: $$\langle (x_1,y_1,z_1),(x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ Let $\Lambda = (\lambda_{ij})$ be an isometry of $\Bbb L^3$. I ...
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1answer
12 views

Show a smooth map from a compact, connected, orientable surface to a cyllinder has singular derivative at 2 points.

Let $M$ be a compact, connected, orientable surface in $\mathbb{R}^3$. Let $N$ be the cyllinder in $\mathbb{R}^3$ defined by $x^2+y^2=1$. Suppose $f:M\to N$ is $C^{\infty}$. Show that $f_*:TM\to ...
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1answer
35 views

Are distance-related paradoxes limited by the size of an atom?

See these 2 paradoxes: Coastline paradox The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. ...
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1answer
35 views

Is there a relation between these words?

I need to know about the difference or relation or are they same: Interpolating geometry; is a system of generators such that given distinct points no two of them have same first coordinates, there ...
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Connection in fibre bundle from discontinuous group action

I am trying to understand connections in fibre bundles. I thought of the following problem: Let $\Gamma$ be the discrete group generated by \begin{pmatrix} 1 & 3 & 0 \\ 0 & 1 & 0 \\ ...
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Can a quaternionic Kähler manifold be NOT Kähler?

I have an explicit construction of the metric on the quaternionic Kähler manifold $$\mathcal M = \frac{Sp(1, 1)}{Sp(1) \times Sp(1)}.$$ Arranging the four real degrees of freedom into two complex ones ...
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1answer
37 views

The definition of scalar and vector concomitant of a metric

I'm reading Defrise-Carter's paper Conformal Groups and Conformally Equivalent Isometry Groups. One might find the paper at the following link: ...
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1answer
55 views

Osculating circles intersecting a given point

Well, the problem is a question in Montiel's book. How to prove that a planar curve $\alpha$ such that all osculating circles intersects a given point is actually a circle (or a part of it)? I've ...
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54 views

Zero set of finitely many polynomials.

Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed ...
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4answers
119 views

Proving $\dfrac{dN}{ds}=-\kappa T+\tau B$

Currently revising for a differential geometry exam. The question I am working on is one of those types where the next part of the question follows from the last. I've gotten to the point where I have ...
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Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
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2answers
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Embedding of $\mathbb{R}^2 \to \mathbb{R}^3$ with non-parallel tangent planes

I have a qual question here and I'm struggling to get a good starting point. The question asks to construct a smooth proper embedding $f\colon \mathbb{R}^2 \to \mathbb{R}^3$ such that for any distinct ...
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1answer
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Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
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1answer
218 views

Christoffel Symbols and the change of transformation law.

I have seen it written that the change of co-ordinate form is given by the following: $$ \tilde \Gamma^{i}_{jk} = {\partial \tilde x^i \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta ...
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1answer
242 views

Restriction of a differentiable map $R^3\rightarrow R^3$ to a regular surface is also differentiable.

This is again an excercise from Do Carmo's book. Prove: if $f:R^3 \rightarrow R^3$ is a linear map and $S \subset R^3$ is a regular surface invariant under $L,$ i.e, $L(S)\subset S$, then the ...
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1answer
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Russian Texts on Geometry

I recently saw a question today pertaining to Russian mathematics and I have a similar question but of a slightly different flavor. I've always heard that the Soviet Union had a history of producing ...
2
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1answer
81 views

Hodge star operator

Again I have issues with notations. The hodge star operator is defined as : (m is the dimension of the manifold) $$\star: \Omega^{r}(M) \rightarrow \Omega^{m-r}(M)$$ $$\star(dx^{\mu_{1}} \wedge ...
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1answer
37 views

Riemannian Metric Notation

I am just being introduced to Riemannian metrics, and I am having a bit of confusion on the notation. When reading, I've encountered some different notation in different sources, so I want to make ...
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Alexandrov embedness and branch points

Let assume that $\Sigma_n$ is a sequence of compact surfaces in $\mathbb{R}^3$ of fixed genus. We assume that the surfaces are Alexandrov embedded, that is to say there exits an immersion $i_n$ from ...
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Finding a lie group structure on $\mathbb R^n\setminus\{0\}$

I want to find all maps $g: \mathbb R^n\setminus \{0\} \rightarrow GL_n(\mathbb R)$ which satisfy the properties $g$ is differentiable and injective $g(g(a)b) = g(a)g(b)$ for all $a,b\in\mathbb ...
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1answer
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Differential Forms on submanifolds

Say I take an embedded submanifold of $\Bbb R^n$, like the sphere. Any differential form on $\Bbb R^n$ can be restricted to the sphere. My question is this: is any differential form on the sphere (or ...
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1answer
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Levi-Cevita symbols: Why is $\epsilon_{ijk}\epsilon_{pjk}$ equal to $2\delta_{ip}$, but not $0$?

I'm learning vector calculus on my own and sometimes strange things happen that I don't know how I should explain them. We have this famous equality: ...
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2answers
87 views

Is a 1-form locally expressible as $dx$?

Given a 1-form $\alpha$ which is non-zero at every point of a manifold $M^n$, is it true that locally I can express the form as $dx$? (that is around each point there is a coordinate neighborhood such ...
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3answers
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maximum area of a rectangle inscribed in a semi - circle with radius r.

A rectangle is inscribed in a semi circle with radius $r$ with one of its sides at the diameter of the semi circle. Find the dimensions of the rectangle so that its area is a maximum. My Try: ...
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An Atlas for $\mathbb{R}/{2\pi \mathbb{Z}} $

I've been having some difficulty finding an atlas for $\mathbb{R}/{2\pi \mathbb{Z}}$. The way I have been thinking of this so far is by using the standard projection map $\pi$ on open intervals of ...
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Orientability of a product of smooth manifolds implies orientability of each factor

I've been learning a bit about orientability on smooth manifolds. I'm having torubles with this exercise: Given two smooth manifolds $M$ and $N$, show that the product manifold $M \times N$ is ...
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Spherical coordinates and Lebesgue integral

I found a textbook that says that $r \in [0,\infty), \theta \in [0,\pi], \phi \in [-\pi,\pi]$ are the spherical coordinates. Then he goes on with: This corresponds to a unitary map $L^2(\mathbb{R}^3) ...
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0answers
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n-form density weight of the levi civita tensor

Why does the n form of levi civita $\epsilon:$=$w_{1}\wedge w_{2} ...\wedge w_{n}$ have a density of weight -1 where $w_{i}$'s are cobasis of some vector basis? I thought this geometrical quantity ...
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0answers
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manifold structure on a finite dimensional real vector space

I'm reading Warner's Foundation of Differential Manifolds and Lie Groups. I don't get how the finite dimensional real vector space gives a natural manifold structure. Someone has asked it before ( ...
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1answer
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How do we check conformal equivalence of parametrized surfaces, e.g. parallel surfaces?

Suppose we have two parametrized surfaces in $\mathbb{R}^3$: $$ X,Y:\mathbb{R}^2 \rightarrow \mathbb{R}^3 $$ The induced metric on either surface is the pullback of the Euclidean metric $\bar g$ due ...
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1answer
60 views

Gradient of norm of embedding

Let $\varphi:(M,g,\nabla)\to\mathbb{R}^n$ be a smooth embedding of a convex hypersurface. I want to explicitly calculate $$\langle \varphi,\varphi_{\ast}(\nabla\|\varphi\|^2)\rangle.$$ In particular, ...
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Differential on a product space as sum of differentials

I would like to derive a rule for differentiating maps of the form $h(f(\bullet),g(\bullet))$ on smooth manifolds which is equivalent to partial differentials of multi-variable functions on $\mathbb ...
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Tangential Vector

I have trouble to understand the following definition. If $M$ is a differentiate manifold then the tangent vector in $p\in M$ is a map $v:\mathcal{F}\rightarrow \mathbb{R}$ such that $v(f)=0$ if ...
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2answers
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Frenet-Serret formula proof

Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$ What is $\tau$, I can't figure that part out. ...
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Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
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3answers
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solenoid and irrotational vector

Let $V$ be a vector point function. $V$ is solenoid if $\operatorname{div} V =0$ and irrotational if $\operatorname{curl} V =0$. How can one visualize examples of solenoid or irrotational functions? ...
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What distinguishes elliptical coordinates from polar coordinates?

I am trying to identify what characteristic distinguishes elliptical coordinates from polar coordinates. For concreteness, let's write down the expressions. Polar: $$ x=r \cos(t) \\ y=r \sin(t) $$ ...
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Law of Sines Non-Euclidean geometry

Is the following Law of Sines valid on all surfaces isometric to a sphere? $$\frac{\sin A }{ \sin a }= \cdots = \frac{ \sin C }{ \sin c } = E.$$ And similarly, Is the following Law of hyperbolic ...
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1answer
69 views

Does a compact manifold require non zero Ricci curvature?

Imagine we have a Riemanian compact manifold. Does the compactness necesarily make its curvature non zero? If the answer were no, does anyone know any such manifold with isometry group ...