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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Surface area of Convex bodies contained in one another

Suppose we have two compact convex bodies one contained in the other in $\mathbb{R}^n$, $C\subset D\subset \mathbb{R}^n$. Does it follow that the ($n-1$ dimensional) surface area of $C$ is less than ...
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How do I show that the reparametrization of a pre-geodesic is pre-geodesic?

So this is probably a very silly question but I need to prove the reparametrization of a pre-geodesic is a pre-geodesic. The hint in my book says it's obvious, so I am clearly missing something.
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basic question of differential geometry.

I'm taking a course on general relativity and I did a problem that I have now found out I have no idea how to interpret. The surface of a paraboloid has the metric $$ds^2=(1+r^2)dr^2+r^2d\theta^2$$ ...
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26 views

Proper patch in the differential geometry

I have a question that coincides with this question. Proving that every patch in a surface $M$ in $R^3$ is proper. Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries ...
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1answer
30 views

Every compact hypersurface in $\mathbb{R}^n$ is orientable

Show that every compact hypersurface in $\mathbb{R}^n$ is orientable. HINT: Jordan-Brouwer Separation Theorem. This is an exercise from Guillemin and Pollack. So hypersurface means smooth ...
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Example of a Symbol: Connection.

I'm trying to get more intuition for the symbol of a differential operator. In particular, I've tried the example of a connection. What is the most efficient way to compute the symbol for, say, a ...
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15 views

Finding a zero of homogeneous parts of polynomials using zero of those polynomials

Let $f, g \in \mathbb{Q}[x_1, ..., x_n]$ be polynomials of degree $d$. Let $F$ and $G$ denote the degree $d$ portions of $f$ and $g$ respectively. Suppose there exists a non-singular point ...
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1answer
45 views

Arc length of implicit curve using gradient magnitude of the unit step function?

I came across a funny formula for the arc length of an implicit curve in the paper here, given at the top of page 5. Let me set the context: Consider a function $\phi: \mathbb{R}^2 \to \mathbb{R}$ ...
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49 views

Differentiating the pull-back of a one-form

Let $\Omega$ be an open subset of a vector space $V$ and let $\alpha\colon \Omega\to V^*$ be a one-form on $\Omega$. Assuming that $\alpha$ is differentiable, then for any $x\in \Omega$, $D\alpha ...
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1answer
27 views

How to rearrange this doubly infinite sum for a diffeomorphism using the existence of a first integral?

Let's take two diffeomorphisms $F,G: \mathbb{R}^{n} \mapsto \mathbb{R}^{n}$. Let $x \in \mathbb{R}^{n}$, and $x^{n} = F^{n}(x)$, where $n \in \mathbb{Z}$. Suppose that $F$ has a first integral, i.e. ...
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21 views

Proving the invariance of the mixed product under direct congruent transformations

Text: Differential Geometry, by Erwin Kreyszig Given a set of vectors: $$ \overrightarrow{a}=\langle a_1, a_2, a_3 \rangle $$ $$ \overrightarrow{b}=\langle b_1, b_2, b_3 \rangle $$ $$ ...
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1answer
39 views

Proof of Euler's Theorem involving curvature.

Theorem: Let $φ$ be the angle, in the tangent plane, measured counterclockwise from the direction of minimum curvature $\kappa_1$ . Then the normal curvature $\kappa_n(φ)$ in direction $φ$ is given by ...
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1answer
28 views

Definition of Levi-Civita connection map?

Does anyone know definition of Levi-Civita connection map that defined as $TTM\to TM$. I would appreciate if you could give a good reference. Thanks.
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29 views

Can the hyperbolic orbifold 2*55 be smoothly and isometrically embedded in 3-space?

Grow a square in the hyperbolic plane until its vertex angles become $\pi/5$. Assuming that the constant Gaussian curvature of our hyperbolic plane is $-1$, the sides of the resulting hyperbolic ...
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2answers
42 views

Exact vs. conservative

I'm having trouble understanding definitions. What's the difference between something being exact and being conservative? I understand both involve proving that a potential function $f$ exists such ...
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18 views

Proof that the change of parameters between two regular surfaces is a diffeomorphism

I'm stuck on the proof that the change of parameters between two regular surfaces is a diffeomorphism. I'm using Do Carmo's book Differential Geometry of Curves and Surfaces, which can be found online ...
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1answer
28 views

Level set of the hopf map

The Hopf map is given by the projection $\pi: \mathbb{S}^3 \to \mathbb{S}^2$, and: $\pi: z \mapsto zi\bar{z}$, where $z \in \mathbb{S}^3$ and $i \in \mathbb{H}$ is a unit quaternion. Show that ...
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1answer
16 views

The bundle of spin frames as an associated bundle

$\DeclareMathOperator{\Spin}{Spin}$Let $X$ be an oriented smooth $n$-manifold with the frame bundle $\pi_{SO} \colon F_{SO} \to X$. Then the bundle of spin frames is a $\Spin(n)$-bundle $\pi_{\Spin} ...
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3answers
62 views

Degree 1 map from torus to sphere

I'm trying to find a smooth degree 1 map from the torus $T^2 = S^1 \times S^1$ to the 2-sphere $S^2$. My first thought was to use the two coordinates $(\theta_1,\theta_2)$ to map onto the usual ...
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0answers
52 views

Exterior derivative for functions with values in a parallelizable manifold

In Sharpe's text on Cartan geometry, he explains in section 1.5 on page 52 how to define an exterior derivative for maps into a parallelizable manifold $N$. Let $f: M \to N$ be a smooth map, and ...
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1answer
32 views

How many distinct flat connections are there on a flat bundle?

Given a flat smooth vector bundle (i.e. with constant transition functions), how many distinct flat connections could we put on it? If the flat connection is not unique, is it unique up to gauge ...
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2answers
79 views

de Rham cohomology on finitely smooth manifolds

In all of the places I've looked, de Rham cohomology is defined on $\mathcal{C}^\infty$ manifolds with $\mathcal{C}^\infty$ differential forms. What about de Rham cohomology on $\mathcal{C}^r$ ...
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36 views

$\beta(s) = \int_0^s B(u)\,du$ is a unit speed curve

I'm teaching myself in this differential geometry book, so please have some patience with me if I ask something obvious. Let $\alpha(s)$ be a unit speed curve with domain $(-\epsilon,\epsilon)$ and ...
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1answer
26 views

Calculate the Euler-Poincaré characteristic of followin surfaces.

Calculate the Euler-Poincaré characteristic of: An ellipsoid. The surfase $S=\left\{ \left(x,y,z\right)\in\mathbb{R}^{3}:x^{2}+y^{10}+z^{6}=1\right\} $. Note: Not how to do this problem, I not ...
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calculation of laplacian- beltrami operator on sphere in terms of (x,y,z)?

As the answer given by @mark fischler in my previous question How to change metric variable to x,y,z The metric on sphere could be written in terms of x,y,z could be written as $(ds)^2$: $$ (ds)^2 ...
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11 views

Change of cut off function

When we have a Dirichlet integral $\int_{B_{0}(R)}|\nabla f|^{2}dv = O(R^{2})$, when $R\to \infty$. And if we use the cut off function $\phi_{R}$ defined as $\phi_{R}=1$ for $x\in B_{e^{R}}$, ...
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1answer
40 views

Why is the Einstein Static Universe an infinite cylinder?

The Einstein static universe metric is $$ds^2=-dt^2 + d\chi^2 + \sin(\chi)^2d\Omega^2$$ where $-\infty<t<\infty$ , $0<\chi<\pi$ and $d\Omega^2$ is the metric on a $S^2$. It describes the ...
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25 views

Discretizations of Differential, Geometric and Topological Notions

I have noticed a recurring theme in Graph Theory / Theoretical Computer Science (abbreviated GT and TCS throughout this post) in that notions typically belonging to differential calculus / geometry / ...
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1answer
11 views

Finding asymptotic curve

For a given surface with parametrization $f(u,v)$, I obtained following diff.equation for asymptotic curve $$v^2k_3^2\mbox{d}u^2+2k_3\mbox{d}u\mbox{d}v=0.$$ Solwing this I got one solution ...
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Apply “thickness” to a minimal surface

By definition a minimal surface has no volume. But my goal is to give the minimal surface some volume. Is there a mathematical way to do this? I found a thread in a forum of a math visualization tool ...
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1answer
21 views

Foliated vector fields span tangent bundle

Suppose we have a foliation on a manifold $M$ which will be call $F$. Foliated vector fields are those $X$ for which: $$[X,T] \in TF$$ for all $T \in TF$. It is easy to see that locally if we have a ...
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1answer
44 views

Differential forms on a point

For the proof of Poincaré lemma, it's essential to evaluate $\Omega^p(*)$ where $*$ is zero dimensional manifold and $\Omega^p$ is a collection of all $p$-forms on given manifold. Clearly, $\Omega^0 ...
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1answer
37 views

Dimension of topology manifold

In the 3 page of Jurgen Jost's Riemannian Geometry and Geometric Analysis .Why it is harder in topology manifold than differentiable manifold ? I think it is easy in differentiable manifold because ...
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1answer
19 views

$S^2 \times S^2$ diffeomorphic to oriented $2$ dimensional vector subspaces of $\mathbb{R}^2$? [duplicate]

As the question title says, is the product of spheres $S^2 \times S^2$ diffeomorphic to the set of oriented $2$ dimensional vector subspaces of $\mathbb{R}^2$?
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1answer
19 views

Parametric Equation of Cycloidal Sine Curve

I am trying to find the parametric equation of a sine curve, which oscillates around a circle as it's $x$-axis. I have done preliminary approximations using Epicycloid parametric equations for the top ...
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1answer
36 views

Intersection of kernels of linearly independent smooth 1-forms on $\mathbb R^n$

I'm trying to solve the following problem: Let $\omega^1,\dots,\omega^k$ be smooth $1$-forms on $\mathbb R^n$ that are linearly independent at each point of $\mathbb R^n$. For $p\in\mathbb R^n$, ...
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19 views

Show that the following statements characterize the straight line as curves in $\mathbb{R}^3$

(1) All tangent lines to the curve pass through a fixed point (2) All tangent lines are pallalel to a given line (3) $k = 0$ ($k$ is the curvature) (4) The curve $\alpha(t)$ satisfies $\alpha'(t)$ ...
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1answer
37 views

An orientation on a $(n-1)$-dimensional submanifold.

This question goes on where this question ended. Given is an non-empty $(n-1)$-dimensional ($n\ge2$) differentiable submanifold $X\subset\mathbb{R}^n$ such that there exists an open ...
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1answer
32 views

Focal point and principal curvature of a surface

Suppose $S$ is a surface parametrized by $f$ and its Gauss map is denoted by $N$. Define a map $f_t(u,v)=f(u,v)+tN(f(u,v))$. Define a focal point $q$ of $S$ as follows: if there is $t\neq 0$ such that ...
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30 views

The map $f:M\rightarrow N$ is differentiable, where $M,N$ are $m$-manifolds.

Suppose the map $f:M\rightarrow N$ is differentiable, where $M,N$ are $m$-manifolds with $M$ compact and $N$ connected. If the degree of $f$ is 1, then $f$ is surjective?
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30 views

Orientability of the level set of a map between abstract oriented manifold

Let M and N be oriented manifold and let $f:M\to N$ be a smooth map between them. Suppose $y \in N$ is a regular value for $f$, how can we show that $f^{-1}(y)$ is orientable? I've seen a solution ...
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2answers
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Curve with all normal planes having a common point

Question: Consider a regular curve parametrized by arclength. If all of the normal planes have a common point then this curve lies on a sphere. I guess this has to do with the Frenet frame, but ...
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1answer
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How to prove that the unit circle in the $x$-$y$ plane is a geodesic on the hyperboloid $x^2+y^2-z^2=1$?

Is there any way (besides a graph) to prove that the unit circle in the $x$-$y$ plane is a geodesic on the hyperboloid $x^2+y^2-z^2=1$?
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1answer
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A theorem of change of variable on submanifolds

I am looking for a proof (or a link to a book or pdf) of the following result : If $\phi : S \to S'$ is a differentiable map between two compact, connected oriented $2$-dimensional submanifolds of ...
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2answers
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Topological boundary of a specific manifold.

Let $\emptyset\neq X\subset\mathbb{R}^n$, $n\ge2$, be an $(n-1)$-dimensional differentiable submanifold, i.e. for every $p\in X$ there is an open $U_p\subset\mathbb{R}^n$ with $p\in U_p$, and a ...
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If $x$ is a fixed point of $f \in Isom(\mathbb{R^2})$, why is $d(x,f(y))=d(x,y)$?

If y $\neq x$ and if $x$ is a fixed point of $f \in Isom(\mathbb{R^2})$, why is $d(x,f(y))=d(x,y)$? I understand that by definition: $d(x,y) = d(f(x),f(y))$ but not the equality above.
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60 views

Does $f^{\ast}$ homotopic to $g^{\ast}$ imply $\int f^{\ast} w = \int g^{\ast} w$?

Let $f,g: M^{k} \to N$ ($M$ and $N$ with out boundary ) such that they are homotopic then for $\omega$ a $k$-form on $N$ do we have that $$ \int_M f^{\ast} \omega = \int_M g^{\ast} \omega$$ as ...
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1answer
45 views

Consider the function : $L_{ij}=x_{i}\frac{\partial}{\partial x_j}-x_{j}\frac{\partial}{\partial x_i}$

Let $\Omega$ be a smooth bounded domain of $\mathbb{S}^n$, the unit $n$ sphere centered at the origin of $\mathbb{R}^{n+1}$, and consider the functions $$L_{ij}u=x_{i}\frac{\partial u}{\partial ...
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21 views

Change of variable : integral on a submanifold

Let $A \in GL_3(\mathbb{R})$. How can I show that $$\int_{\mathbb{S}^2} \frac{dp}{\|Ap\|^3} = \frac{4 \pi}{|det A|}$$ ? My first and only taught is to apply a method of change of variable : $q= ...
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1answer
36 views

Normal vector field to this hypersurface

Let $M^n$ be a hypersurface of the unit sphere $S^{n+1} \subset \mathbb{R}^{n+2}$ contained in the open upper hemisphere $S^{n+1}_+$, and let $N : M \to \mathbb{R}^{n+2}$ be a unit normal vector field ...