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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Meaning of alternation in definition of wedge product

Spivak defines the wedge product as $\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$ and I have been running into some conceptual issues here. The alternation is defined as ...
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Mobius strip as manifold and as a bundle over $S^1$

I am trying to construct the Mobius strip bundle onver $S^1$. I am stuck in how to define charts for the Mobius bundle. Once I make this, the problem will easily be solved. My attempt was: $$M = ...
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23 views

On a parallelizable manifold, is there always a frame satisfying $[X_i,X_j]=0$?

Let $M$ be a parallelizable manifold. Is there always a global frame $(X_i)$ such that $[X_i,X_j]=0$ for all $i,j$ ? If the answer is no, what kind of obstruction there is to find such a frame ? ...
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31 views

Two definitions of the first Chern class

there are two definitions of the first Chern class that I don't know how to relate - hints and references are both welcome. So, first approach: say that I have a complex vector bundle $E\to M$; I can ...
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22 views

Normal vector field for an immersion

I know that a hypersurface $M$ of a riemannian manifold $N$ is orientable iff there exists a globally defined unit normal vector field $\eta : M \to TM^{\perp} \subset TN|_M$. Does the same hold for ...
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On Automorphism on Space of Smooth Functions

Given a Operator $T$ (an Automorphism) on the subspace $X$ of Smooth functions on $\mathbb R ^n$, $\mathcal C^\infty(\mathbb R^n)$ $$ X=\{u \in \mathcal C^\infty (\mathbb R^n) \,|\, supp ...
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Metric of the doubling i.e circular billard table

Let $D^2=B_1(0) \cup_{\partial B_1(0)} B_1(0)$ denote the doubling in $\mathbb{R}^2$ i.e. the metric space which one gets after gluing two closed balls of radius $1$ with their induced metric coming ...
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parameterization of surface of revolution?

parameterization of surface of revolution formed by revolving the $x=\cosh z$ around z axis , i thought the it as $$x=\cosh z \cos \theta ,y=\cosh z \sin \theta ,z=z$$ Hence the surface can be ...
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Showing that a mapping is an isometry

Can anyone please help with this question? I have tried substituting but I'm not sure if that's correct so think I am missing something. Let S denote the surface of revolution $(x, y, z) = ...
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22 views

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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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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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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24 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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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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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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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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what does it mean to have inner product of $S^2$ and $R^3$?

It may be that the title of my question is wrong but i am writing this question because i am struck while reading this paper Brownian motion on rotational group Where $^*\mathscr{f} $ is transpose of ...
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Zero gradient in $L^2(M)$

I'd like to show that for $u \in L^2(M)$, for M a compact, connected Riemannian manifold, if $\nabla_g u = 0$ (i.e $\forall X$ $C^{\infty}$- vector field on $M$, $\int_M u \hspace{1mm} \text{div}_g X ...
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25 views

Can the rank of harmonic map decrease far from the boundary?

This question is in some sense a continuation of this question, though it asks something weaker. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are ...
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Is this proof about the Lie brackets and flows correct as given?

In this post, Fredrik Meyer gives a proof to the following formula(Please see the conditions and the meaning of notations in the link): $\frac{d}{dt}|_{t=0} \alpha(t) = [X,Y](p)$, where ...
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69 views

Explicit form of the exponential map

I am stuck by the following problem. Let $ g = 1/y^2 (dx \otimes dx + dy \otimes dy)$ be a Riemannian metric on the half-plane $ S = \mathbb{R}^2_{y > 0}$. What is the explicit form of the ...
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26 views

On lifts of a trajectory of a quadratic differential

Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. The differential $q$ defines a flat metric with conical singularities on $X$: if $q=f(z)dz^2$ ...
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22 views

A space curve with non-vanishing curvature is planar iff its torsion is 0

Intuitively this is simple and to prove the backwards direction: $\tau = 0 \Rightarrow \mathbf{b'}=0 \Rightarrow \mathbf{b} $ is constant. Then letting $\gamma$ be the parametrisation of the curve ...
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a algebraic concept in reimannian geometry?

let $(M,g)$ is a $C^{\infty}$ manifold with product form of $M=R \times \Theta $ the for each element $\varepsilon (r,\theta )\in M$, the tangent space $T_{\varepsilon}M$ is natural isomorphic to ...
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Exercise concerning areas inside closed curves

Let $\alpha (s)$, $s\in[0,l]$, be a closed, convex, plane curve with $\kappa >0$. Let $r$ be a positive constant and define $\beta (s)=\alpha (s)-rn(s)$, where $n(s)$ is the normal vector of ...
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18 views

Height function of a hypersurface

I was reading an article by do Carmo and Warner, which says: "By the height function for an oriented hypersurface at a point $p$ we shall mean the function defined on a neighborhood of the origin in ...
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45 views

integral of a vector field in $\mathbb{R}^n$

I'm wondering the definition of the integral of a vector field on a hypersurface in R^n. Here is what I guess, but I did not found it on the internet. Let $v$ be a vector field on $\mathbb{R}^n$ and ...
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37 views

Sobolev spaces on Riemannian manifold

I know one can define the Sobolev spaces on a Riemannian manifold as completions, but is there an equivalent definition that uses weak derivatives, like in the case of open sets in $\mathbb{R}^n$ ? ...
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27 views

Is there a mathematical treatment of continuously (or smoothly) deforming surfaces, or, in general, continuously deforming manifolds?

I'm wondering if there's been any attempts to construct formal mathematical descriptions of deforming surfaces, whether evolving to some steady state configuration or reacting to some pulse, force, or ...
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Connected components of the complement of a geodesic

I came across the book "Knots, Molecules, and the Universe: An Introduction to Topology" by E. Flapan which is quite nice. In the first chapters it is discussed how one can distinguish the sphere ...
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A piecewise regular simple closed curve bisects the area of the unit sphere if and only if it has total geodesic curvature 0

How can I prove that "A piecewise regular simple closed curve bisects (this curve splits the unit sphere into two pieces, the area of which are equal) the area of the unit sphere if and only if it has ...
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22 views

Gaussian curvature in polar coordinates

Find the expression for the Gauss curvature in the polar coordinates associated to the exponential map. I thought about using Gauss's lemma: if $(r,\theta)$ are polar coordinates in the tangent plane ...
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36 views

How to show $\nabla_a\nabla_c R_{ad}-\nabla_c \nabla _a R_{ad}=-R_{cade} R_{ae}-R_{ce} R _{de}$?

$\nabla$ is Riemann connection and $R_{ij}=g^{kl}R_{ikjl}$. How to show $\nabla_a\nabla_c R_{ad}-\nabla_c \nabla _a R_{ad}=-R_{cade} R_{ae}-R_{ce} R _{de}$ ? Or generate commutator of generate ...
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Why do we need the $S \otimes L^{1/2}$ bundle product to determine a $Spin_c$ structure?

I am reading Marino's book on topological field theory and 4-manifolds and I am very confused in the construction of the $Spin_c$ structures for manifolds that do not admit a $Spin$ structure. In the ...
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21 views

Construct a lifting from the the total space of a fiber bundle to its spoace of 1-jets

I am reading the book "Convex Integrtion Theory by D.Spring" and need some help. Let $\pi:X \rightarrow V$ be a smooth fiber bundle over a smooth base manifold $V$. Let $h\in\Gamma^{0}(X)$ and ...
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finding geodesics on the surface $z=x^2$

Find all the geodesics on the surface $z=x^2$. I found the metric and the Christoffel symbols but i do not know what to do next, any hint ?
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symmetric and alternating tensors in differential geometry

The following is an excerpt from Chern's Lectures on Differential Geometry: I don't see how the proof shows the other direction of the set inclusion. Would anybody explain the logic in the ...
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In proof of tangent space being a plane

can someone explain why this is true? If $\sigma$ is a surface patch of a surface $S$ and $p$ is a point on the image of $\sigma$ and if $p$ lies in the image of a curve $\gamma$ contained in $S$ say ...
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14 views

Property on the derivative of a wedge product of two n-forms

I'm trying to prove the following property of $n$-forms. When $w_1$ is a $n_1$-form on $M$, $w_2$ a $n_2$-form also on $M$, and $d$ denotes the exterior derivative $$\require{cancel} d(w_1\wedge ...
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28 views

Dirichlet problem for a ball in a Riemannian Manifold

I'm trying to work through the 2nd chapter "Volume comparison" of Jeff Cheeger's "Degeneration of Riemannian Metrics under Ricci Curvature Bounds". But with the last section I have some problems. ...
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Confusion regarding terminology in Pressley's E.D.G

Here are two definitions taken from page $77$ of Pressley's Elementary Differential Geometry - $2$nd edition. Definition $4.2.1$ A surface patch $\sigma: U \to \Bbb R^3$ is called ...
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23 views

Metric evolving under Ricci flow with nonnegative scalar curvature is shrinking?

Let $g_{ij}(x,t)$ be a complete solution of the Ricci flow on a surface with bounded and nonnegative scalar curvature ,why the metric is shrinking ?
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What is $P\times_G E$?

I know what is principal bundle and associated bundle according Wiki.But I am not understand what is $P\times_G E$ .Seemly it is bundle,but I am not sure what structure is it . Below picture is from ...
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Eigenfunction basis of Laplacian on a manifold

It is a well known result that for $\Omega$ bounded open set in $\mathbb{R}^n$, there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(\Omega)$. It is also known that there exists ...
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Why is this proof that a circular cone is not a surface not rigorous?

In example $4.1.5$, page $73$ of Pressley's Elementary Differential Geometry, a "heuristic" argument is given to prove that the circular cone with vertex the origin and angle $\pi/4$, is not a ...
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What is skew-product decomposition?

What is skew-product decomposition of Brownian motions referring to this paper Pauwels ,Rogers
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Equivariant flat U(1) bundle on a torus

I am trying to understand equivariant flat $U(1)$ bundles on a torus, $(S^1)^N$. By equivariant, I mean equivariance with respect to the natural action of $(U(1))^N$ on $(S^1)^N$. $G$-equivariant ...
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When and where to check the formal definition of a manifold

In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ...
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39 views

An intuitive question about the metric $d(\cdot,\cdot)$ on a complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian manifold. Suppose $\gamma_1,\gamma_2:[0,1] \to M$ are two different segments(i.e.minimizing geodesics) from $p$ to $x=\exp_p(v)$. Suppose further that ...
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Confusion surrounding the Koszul-Malgrange theorem

On a recent project, I ran into something which seemed related to the Koszul-Malgrange theorem. According to nlab, the original statement of the theorem is Theorem 1. (Koszul-Malgrange theorem) ...