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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When can we write evolving curves as curves over a fixed curve?

Suppose $\gamma(t)$ for each $t$ is a curve. We may write $$\gamma(t)(s) = \gamma_0(s) + d(t,s)N(s)$$ where $\gamma_0$ is some fixed curve, $N$ is the unit normal vector and $d$ is a distance from ...
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58 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 ...
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240 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 ...
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68 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 ...
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96 views

Manifold contains a totally geodesic closed hypersurface

Let $(M^n,g)$ be a closed simply-connected positively curved manifold. Show that if $M$ contains a totally geodesic closed hypersurface (i.e., the second fndamental form or shape operator is zero), ...
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149 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 ...
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97 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 ...
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143 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 ...
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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 ...
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316 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 ...
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321 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 $x = x_{0}t + \sqrt{1-t^2}y$, ...
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178 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 ...
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106 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 ...
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118 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 ...
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231 views

The connected Lie subgroups of a n-dimensional torus.

I am stuck on the following question Let $T^n$ be a $n$-dimensional torus. What are the connected Lie subgroups of $T^n$? Which of these are closed subgroups? Does anyone have any ideas on how to ...
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128 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 $R[f] \colon \mathcal{H} \to \mathbb{R}$ ...
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217 views

Extending Tensor Fields defined on Manifolds to Ambient Space

I am currently reading about tensor fields on manifolds, and I came across two comments that sound contradictory to me. The first comment is made in the book by James Munkres "Analysis on Manifolds", ...
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171 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
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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, ...
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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 ...
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78 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 ...
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133 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 ...
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585 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 ...
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135 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 ...
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196 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 ...
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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" ...
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279 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 ...
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173 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 ...
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179 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 ...
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142 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 $H^{1}(X,\mathbb Z)^*$. ...
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209 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 ...
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289 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 ...
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124 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 ...
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121 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
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135 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 ...
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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 ...
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156 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 ...
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20 views

Every non-constant closed curve has positive period

I want to show that every non-constant closed curve has positive period, but i'm not really sure how to do this. A smooth curve $r(t): \mathbb{R} \to \mathbb{R}^n$ is $T$-periodic if $r(t+T)= r(t)$ ...
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24 views

moving frame with maple

I have already ask this question on stackoverflow, but since it concerns as mathematics than computer science, I ask it here too. I would like to make a classical computation using maple. I would ...
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19 views

geometric meaning of conjugate points

Recently I am reading Manfredo do Carmo's Differential Geometry of Curves and Surfaces. He said the $q$ is the conjugate point of $p$ with respect to a geodesic $\gamma$ joining the two points if ...
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38 views

Equivariant generalization of $\mathcal{O}(1)$ on $\mathbb{CP}^1$

The connection of the line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^1$ is given by \begin{equation} A=\frac{i}{2}\frac{\overline{z} \, dz-z\,d\overline{z}}{1+|z|^2} \end{equation} This follows since ...
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0answers
22 views

Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: ...
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19 views

Upper bound on hessian

Given a smooth Riemannian manifold $(\mathcal{M},g)$ and $f \in C^{\infty}(\mathcal{M})$ let $r(x)= d(x,x_0)$ where $d$ is the distance function wrt $g$ and $x_0$ is some point on the manifold. If we ...
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27 views

What is an integral differential form and how do we recognize it as such?

I am reading about embedding theorems of various types of manifolds (Kodaira's embedding theorem being one of them), and one condition that repeats in all of them is that the manifolds should be ...
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13 views

Regarding the axis of screw motion for a space curve.

The axis of the accompanying screw motion of a curve $c(s)$ at any point $c(s_0)$ is the line in the direction of the Darboux vector $\tau(s_0) T(s_0) + \kappa(s_0)B(s_0),$ through the point $$P(s_0) ...
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30 views

Extending automorphisms on surfaces

Assume we are in the complex setting. Let $X$ be a surface, $C$ a curve on $X$. Say $X-C$ is isomorphic to some $X'-C'$ whith $X'$ a surface and $C'$ a curve on $X'$. If it helps we may assume that ...
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26 views

Why is the backwards curve needed for the inverse of a parallel transport?

Dearh math.stackeschange-community, I'm at a loss with the following problem: Let $I=(a,b)$, then the inverse of the parallel transport $P_{s,t}^\gamma$ from $s \in I$ to $t \in I$ along the curve ...
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16 views

Showing that $(\mathbb{R}, \mathscr{F})$ and $(\mathbb{R},\mathscr{F_1})$ are diffeomorphic but $\mathscr{F}\neq \mathscr{F_1}$

Background $M$ is locally Euclidean with dimension $d$ if $M$ is hausdorff and every point in $M$ has a neighborhood homeomorphic to $\mathbb{R{^d}}$. If $U\subset M$ is open and connected and $\phi$ ...
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27 views

derivative of a projection matrix

The projection onto a parametrised vector $v(\lambda)$ is $P_v = \frac{vv^{T}}{v^{T}v}.$ Its complement is $$P = I-\frac{vv^T}{v^{T}v}.$$ I've got an expression containing this complementary ...
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22 views

Hessian of the Stereographic projection

Consider the stereographic projection from the sphere $S^n$ onto $\mathbb{R}^n$, and take the usual local spherical (polar) coordinates $\omega_1,..,\omega_n$ on $S^n$ (coming from its embedding in ...