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)

2
votes
0answers
61 views

Calculation of |[X,Y]^V|

I want to follow the proof of Theorem 3.1 in "On Eschenburg's Habilitation on Biquotients" - Wolfgang Ziller. The situation is as follows: $Q$ is a biinvariant metric on $G$. So from the ...
2
votes
0answers
241 views

Closed Geodesics as minimisers of action functional

Suppose I have a Riemannian surface $(M,g)$. It's clear that closed geodesics are critical points of the length functional $l(\gamma)=\int\left|\gamma(t)^{\prime}\right|_{g(\gamma(t))}dt$ over the ...
2
votes
0answers
69 views

Unit partition to produce smooth function from continuous ones

Given a positive continuous function (except on closed set, where is zero ) on a smooth manifold how to find a smooth function under the same conditions being less (or equal) than this one ...
2
votes
0answers
158 views

A series of Lemmas about $C^{\infty}$ vector fields

Warner page-37: Theorem: Let $X$ be a $C^{\infty}$ vector field on a differentiable manifold $M$ For each $m\in M$ there exist $a(m)$ and $b(m)$ in extended real line, and smooth curve ...
2
votes
0answers
98 views

flows on a manifold and liebracket

I have the following question: Let $M$ be a smooth manifold and let $p \in M$. Furthermore let $X$ and $Y$ be two vector fields in a neighbourhood $U$ of $p$ and consider their flows ...
2
votes
0answers
148 views

Smooth deformation retracts

Under what circumstances can it be concluded that if two items from the smooth category are related by a topological relationship, then they are also smoothly related in the corresponding way? For ...
2
votes
0answers
46 views

On what quantities is a (differential geometric) connection not allowed to depend?

If we parallel transport a vector $v$ along a closed curve $c_1$, the vector will end up as $v'=A_{c_1}v$, where $A_{c_1}$ is an element of the holonomy group. If we consider composition of curves ...
2
votes
0answers
327 views

De Rham cohomology of the euclidean space without n lines

How can I compute the de Rham cohomology of $\mathbb{R}^3$ minus n lines through the origin? I would like to do it with the Mayer-Vietoris sequence (which is the only thing I know to calculate ...
2
votes
0answers
332 views

Change of variables formula

Consider an example. Let $f(x)$ be a function on the unit sphere $S^{n-1}$. Consider an integral $$ \int\limits_{S^{n-1}} f(x) \, dx $$ I want to make a substitution $x = x_{0}t + \sqrt{1-t^2}y$, ...
2
votes
0answers
181 views

Isometry from a surface to itself.

We are talking about surfaces in $\mathbb{R}^3$ here. I know that not every isometry from a surface to another is a congruence. But what about isometries from a surface to itself? Can someone give ...
2
votes
0answers
108 views

Different definitions of the complex Grassmannian

I have come across two different definitions of what I suspect is the same object. Both are called the complex Grassmannian: 1: $U(n)/U(k)\times U(n-k)$ 2: $SU(n)/(S(U(k)\times U(n-k))$ What is the ...
2
votes
0answers
122 views

Help with this geometric PDE weak formulation and solution

Suppose $X:A \subset \mathbb{R}^n \to \mathbb{R}^{n+1}$ is a parametrisation of a surface $\Gamma$ such that $X(\theta_1, \theta_2) = (\theta_1, \theta_2, h(\theta_1, \theta_2))$ (i.e., $\Gamma$ is a ...
2
votes
0answers
129 views

Different proofs of support theorem for Radon Transform

Let $\mathcal{H}$ be a set of all hyperplanes in $\mathbb{R}^n$. Radon transform of function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ is defined as function $R[f] \colon \mathcal{H} \to \mathbb{R}$ ...
2
votes
0answers
177 views

Reversing a roulette on a straight line - solving for a parameterization?

(See below for update.) I would like to reverse the following equations. Here, the path is traced out by the polar origin of the wheel. You can put the trace point anywhere on or off of the wheel ...
2
votes
0answers
54 views

Deriving PDE to evolve a planar curve to a circle, while preserving its length.

I would like to ask, how could I derive a PDE that evolves a planar curve to a circle while preserving its length ? (The problem states that there is no need to show that the steady state is a circle, ...
2
votes
0answers
130 views

Determinant of the Laplacian of a surface is this correct?

given a surface with metric $ g_{ab} $ i would like to evaluate the functional determinant of the Laplacian in the form $ - \partial _{s} \zeta (0,E^{2})=\log\det( \Delta + E^{2}) $ then i need to ...
2
votes
0answers
79 views

A lower positive bound on the number of closed orbits with given energy for a mechanical system

Let be given a mechanical system with configuration manifold $M,$ potential energy $V$ and kinetic energy $K$ corresponding to a riemannian metric on $M.$ Its dynamics is determined by the ...
2
votes
0answers
136 views

Unified Definition of a Surface

As I noted in this question, there's a lot of inconsistent terminology in use with regard to "curves", "smooth curves" etc and similar comments could apply to the definition of a surface. To get all ...
2
votes
0answers
621 views

verify exercise parallel transport on sphere

I need to solve the following exercise. I wonder whether my solution is correct. Problem: Take a sphere in $\mathbb{R}^3$ centered around the origin of radius $R$. Consider the spherical triangle ...
2
votes
0answers
143 views

A question on generalized Gauss-Bonnet theorem

I am trying to learn about basic characteristic classes and Generalized Gauss-Bonnet Theorem, and my main reference at the moment is From Calculus to Cohomology by Madsen & Tornehave. I know the ...
2
votes
0answers
199 views

How does a geodesic equation on an n-manifold deal with singularities?

My general premise is that I want to investigate the transformations between two distinct sets of vertices on n-dimensional manifolds by: Minimalizing the change in the fundamental shape of the ...
2
votes
0answers
44 views

Evaluating the “regularity” of a mapping $\mathbb{R}^2 \rightarrow \mathbb{R}^2$

Let $R \subset \mathbb{R}^2$ and $R' \subset \mathbb{R}^2$ be two regions in the plane, and $F: R \rightarrow R'$ a smooth map. I would like to find a reasonable measure of the "regularity" ...
2
votes
0answers
298 views

How to construct a vector field without zero on an open manifold?

a friend asked me to pose the following problem: It is known that on an open manifold (connected, not compact and without boundary) there exists a vector field without zero, since its Euler ...
2
votes
0answers
179 views

Curvature of a Connection of vector bundle

Let $X$ be a scheme or manifold and $\nabla: V \rightarrow \Omega^1 \otimes V$ be a connection on a vector bundle $V$ on $X$. Let $R:=\nabla^2$ denote the curvature homomorphism. Does it hold that ...
2
votes
0answers
189 views

Covariant derivative of composition of two tensors

Suppose $TM\to M$ is the tengent bundle over the close Riemannian manifold $M$. Let $\nabla$ be the Levi-Civita connection, $S$ and $T$ are two $(1,1)$-tensor, i.e. at each point $x\in M$, we can view ...
2
votes
0answers
145 views

Complex torus and integral of forms

Start with a complex torus $X$ of dimension $n$ and a basis $e_1,...,e_{2n}$ for its first integral cohomology. Let $f_1,...f_{2n}$ be the dual basis vectors, so they are in $H^{1}(X,\mathbb Z)^*$. ...
2
votes
0answers
213 views

On a remark in Foundations of mechanics, 2nd Edition, by Abraham and Marsden

Given a $2$-form $\omega$ on a manifold $M$, let us denote by $N$ the kernel of $\omega$, i.e. $N:=\{u\in TM : \omega(u,\cdot)=0\}$. Their Proposition 5.1.2 shows that if $\omega$ has constant rank ...
2
votes
0answers
302 views

fundamental group of closed surfaces as CW complexes

Let $T$ denote the $2$-torus, $P$ the projective plane, and $nT$/$nP$ the connected sum of $n$ tori/$n$ projective planes respectively. 1) how can I prove, that $nT$ and $nP$ are homeomorphic to ...
2
votes
0answers
433 views

a question about the geodesic circle

How to show that the geodesic circles have constant geodesic curvature on a surface of constant curvature? Thanks
2
votes
0answers
131 views

Question on the transversality between sections

Let $M^n$ be a differentiable manifold and $\pi\colon E\to M$ is $n$-dimensional vector bundle over $M$. We have a zero section $s\colon M\to E$ of $\pi$. How can I make a section $s'$ which is ...
2
votes
0answers
124 views

Question on the transversality

Let $f\colon N^n\to M^{2n}$ be an immersion. Then, we can extend $f$ to $\bar{f}\colon E(\nu_g)\to M$ of the total space of the normal bundle. Let $s_0\colon N\to E(\nu_g)$ be a zero section and ...
2
votes
0answers
137 views

Given a null surface, calculate the manifold it resides in

This problem is related to General Relativity and specifically Black Holes. The manifold is a 4-dimensional space-time with a Minkowski inner product (i.e. if $||v|| = 0$, $v$ is not necessarily ...
2
votes
0answers
124 views

Transformation induced by a spherical mirror

This is at heart a mathematical problem, but is best motivated in physical terms. I'll introduce a very special case and move on to the general case later. Special case An object, taken for ...
2
votes
0answers
157 views

Special types of Sasaki manifolds

i have a question to special cases of Sasaki-manifolds. Let $(M, g, \xi, \eta, \Phi)$ a Sasaki-manifold. In special case maybe $M=S^{2m+1} \cong \mathbb{C}^{m+1}$. This is a Sasaki manifold. But what ...
1
vote
0answers
15 views

When is a stable domain in a minimal surface area minimizing?

A stable domain $D$ in a minimal surface $S\subset \mathbb{R}^3$ is a domain for which the area-functional $A(t):=\int_{S_t}dS_t$ has non-negative second derivative, i.e. $A''(0)\geq 0$, for all ...
1
vote
0answers
15 views

Finding braches of equilibria

Consider the system of two equations in three variables $(x,y,z)$: $$\left\{ \begin{array}{rl} x+y+z+x^7+y^7+z^9 &= 0\\ x-y+z+1-\cos z &=0 \end{array}\right.$$ The point $x^* = ...
1
vote
0answers
17 views

Gradient and Laplacian of a submanifold

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$. Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. Similarly ...
1
vote
0answers
18 views

Mean curvature submanifold

Consider $S^{N-1}$ the unit sphere and let us focus our attention on the cap $$ G=S^{N-1}\cap\{x_N>0\} $$ with boundary $\partial G= S^{N-2}\times\{0\}$: it is quite obvious to see that $G$ is a ...
1
vote
0answers
16 views

Necessary condition for local existence of solution for system of two first order PDEs

Let $f,g$ be two smooth functions on $\mathbb{R}^2$ and $U,V$ be two smooth vector fields on $\mathbb{R}^2$. What is the sufficient condition for local existence of a solution $\phi$ of equations ...
1
vote
0answers
24 views

Frenet frame, curvature and torsion

Given a curve $\gamma:(-1,1)\to\mathbb R^3$ via $$\gamma(t):=(\frac{1}{3}(1+t)^\frac{3}{2},\frac{1}{3}(1-t)^\frac{3}{2},\frac{t}{\sqrt 2})$$ how can I find a) its Frenet frame, curvature and torsion? ...
1
vote
0answers
25 views

Apply flow of $V$ to a segment of a curve, Do you get covariant derivative?

Apply flow of $V$ to a segment of a curve, look at the velocity of the resulting new curve, a perturbation of the original curve. How is the new velocity (tangent) related to the original one (right ...
1
vote
0answers
35 views

Riemannian metric as an operator

In the article http://www.sciencedirect.com/science/article/pii/0370269379905896 authors consider principal bundle $P(M, G)$ and then define induced metric $g$ on $\eta = Sp(A)/G$, where $Sp(A)$ - ...
1
vote
0answers
31 views

1-form that compute the area of parallelogram

Find a form on $\mathbb{R}^4$ that compute the area of parallelogram generate by any pair of vectors $\vec{a}, \vec{b}$ which are in the plane $\pi=\{\vec{x}\in\mathbb{R}^4| ...
1
vote
0answers
24 views

real analytic functions on manifold

Let $M$ be a real analytic manifold of dimension $k$. Is it then always possible to find real analytic functions $f_1, \dots, f_k \colon M \to \mathbb{R}$, such that they are functionally independent ...
1
vote
0answers
20 views

Regular values of $g(x,y)= x^2 - y^2$

I am doing some very introductory studying about manifolds. I wanted to check I was getting the right end of the stick through this example. Could anyone verify/correct my solution to the following ...
1
vote
0answers
49 views

Exterior derivative on principal bundle

In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...
1
vote
0answers
29 views

geodesic flow is proper action

Good evening to everyone. I'm having a problem in the following setting: If I'm having a homogeneous manifold $M=G/K$, where $K \subset G$ is a closed subgroup, I can always find a $G$-invariant ...
1
vote
0answers
16 views

Integral curves of time dependent derivations

Question: Given smooth manifold $M$, with algebra of smooth functions deoted by $C(M)$ let $D_t$ be a time-dependent derivation of $C(M).$ Let $\hat{D}$ be a derivation of $C(M\times \mathbb{R})$ ...
1
vote
0answers
33 views

Defining a differential for quotients

Let $f \colon M \to N$ be a smooth map between smooth manifolds and $f$ being a surjective submersion. Assuming we have a proper Lie-group action $G$ on $M$, with only one orbit type and $G$ acts on ...
1
vote
0answers
45 views

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 ...