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 ...

learn more… | top users | synonyms (1)

1
vote
0answers
25 views

Proving the invariance of the mixed product under direct congruent transformations

Text: Differential Geometry, by Erwin Kreyszig Given a set of vectors: $$ \overrightarrow{a}=\langle a_1, a_2, a_3 \rangle $$ $$ \overrightarrow{b}=\langle b_1, b_2, b_3 \rangle $$ $$ \overrightarrow{...
1
vote
0answers
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 ...
1
vote
0answers
40 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 / ...
1
vote
0answers
16 views

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 ...
1
vote
0answers
31 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?
1
vote
0answers
61 views

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 ...
1
vote
0answers
45 views

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 =...
1
vote
0answers
26 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 ...
1
vote
0answers
19 views

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 $\alpha(...
1
vote
0answers
74 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 ...
1
vote
0answers
28 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 $q=f(z)dz^2$ ...
1
vote
0answers
25 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 ...
1
vote
0answers
24 views

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 $...
1
vote
0answers
11 views

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 $\...
1
vote
0answers
19 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 ...
1
vote
0answers
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 ...
1
vote
0answers
39 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 $\mathbb{R}^n$ ? ...
1
vote
0answers
28 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 ...
1
vote
0answers
19 views

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 $S^2$...
1
vote
0answers
15 views

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 ...
1
vote
0answers
24 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 ...
1
vote
0answers
37 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 ...
1
vote
0answers
16 views

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 ...
1
vote
0answers
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 $\phi\...
1
vote
0answers
27 views

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 ?
1
vote
0answers
36 views

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 "...
1
vote
0answers
20 views

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 ...
1
vote
0answers
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 w_2)=...
1
vote
0answers
31 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. ...
1
vote
0answers
16 views

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 ...
1
vote
0answers
25 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 ?
1
vote
0answers
27 views

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 ...
1
vote
0answers
32 views

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 ...
1
vote
0answers
40 views

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 surface....
1
vote
0answers
13 views

What is skew-product decomposition?

What is skew-product decomposition of Brownian motions referring to this paper Pauwels ,Rogers
1
vote
0answers
55 views

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 ...
1
vote
0answers
32 views

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. ...
1
vote
0answers
41 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 $\gamma'_1(1)...
1
vote
0answers
23 views

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) ...
1
vote
0answers
8 views

Cokernal of the map $B_1 \times B_2 \rightarrow [B_1,B_2]$

I am reading Nakajima's book, Lectures on Hilbert Schemes of Points on Surfaces. In the proof of Theorem 1.9, it needs the following linear algebra fact. Suppose $B_1, B_2$ are two $n \times n$ ...
1
vote
0answers
33 views

Length of a differentiable curve with respect to a Riemannian metric.

Let $X$ be an $n$-dimensional differentiable manifold ($n\ge1$). A Riemannian metric in $X$ is a family $\{g_p\,|\,p\in X\}$, where for all $p\in X$: $g_p:T_pX\times T_pX\to\mathbb{R}$ is an inner ...
1
vote
0answers
50 views

Can you comb the hair on a 4-dimensional coconut?

It is well-known that you can't equip the surface of the unit sphere with a singularity-free coordinate system. Physicists have called this theorem (which is important for the theory of black holes) "...
1
vote
0answers
9 views

Sum two nearest function of two class are the nearest function of the sum class

Suppose $x,\mu:[0,1]\rightarrow \mathbb{R^2}$ two smooth function and $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (0) = 0, \gamma (1) = 1, \gamma$ is a diffeomorphism $\}$. Here $\Gamma$ ...
1
vote
0answers
40 views

A question about Levi-Civita connection and curvature over 3 manifold

Give a 3-manifold M and Riemannian metric $g$, denote $A$ as the Levi-Civita connection on 3-manifold M corresponds to the metric $g$. Denote the curvature of $A$ as $F_A$, choose three bases ${e_1,...
1
vote
0answers
19 views

Discrepancy between line integral over scalar field and line integral over vector field

There is a discrepancy between the line integral over a scalar field and the line integral over a vector field that is bothering me: Say $\gamma$ is a smooth curve. If $\gamma : \mathbb R\to \mathbb ...
1
vote
0answers
29 views

Show that the tangent bundle is locally a product,i.e., $TU=U\times\mathbb{R}^{n}$.

Show that the tangent bundle is locally a product,i.e., $TU=U\times\mathbb{R}^{n}$. Where $TM$ is the set of pairs $(q,v),q\in M,v\in T_{q}M$, if $(U,x)$ is a coordinate system in $M$, then all ...
1
vote
0answers
23 views

Simple singularities of a vector field

Let $M^n \subset \mathbb{R}^{n+1}$ a compact and orientable hypersurface of even dimension, with Gauss map $\gamma : M \to S^n$, that is, $\gamma(p)$ is normal to $M$ at $p \in M$. Let $a, b \in S^n$ ...
1
vote
0answers
44 views

Integral Curves of Gradient-like Vector Fields

If $X$ is a gradient-like vector field of a Morse function $f\colon M\to \mathbb{R}$, then the integral curve $c_p(t)$ starting at an arbitrary point $p$ approaches critical points as $t\to \pm \...
1
vote
0answers
15 views

Need help understanding this example of a distribution

Consider the following example of a distribution (given here): I tried to draw this. If $p=(a,b,c)$ then $$ X_p = (1,0,-b), Y_p = (0,1,0)$$ Then the planes in the distribution are planes spanned ...
1
vote
0answers
24 views

Integration over manifold

Let $M$ be a smooth 2-manifold in $\mathbb{R}^3$ such that $$4x^2+y^2+4z^2 = 4, y \ge 0$$ The boundary of $M$ is the set of points where $$x^2 + z^2 = 1, y = 0$$ Let $\alpha(u,v) = (u,2\sqrt{(1-u^2-...