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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9
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1answer
199 views

Geometric interpretation of reduction of structure group to $SU(n)$.

Let $E \to X$ be a complex vector bundle of rank $k$. Then the structure group of $E$ can be reduced to $U(k)$, as this is equivalent to specifying a hermitian inner product on $E$ which can always ...
9
votes
2answers
231 views

The Wronskian of holomorphic differentials as a q-differential

I just went through a proof of the counting of Weierstrass points on a Riemann surface (References: Reyssat, Quelques aspects des surfaces de Riemann and Farkas & Kra, Riemann Surfaces) that says ...
6
votes
1answer
176 views

Is there a fundamental misunderstanding here or have I made an algebraic slip?

Is there a fundamental misunderstanding here or have I made an algebraic slip? I have a Riemannian metric of the form $ds^2={du^2+dv^2\over 1-u^2-v^2}$ on an open disc and I want to prove that radial ...
3
votes
1answer
533 views

Laplace-Beltrami operator for curves

I'm CS major and have used discrete Laplace-Beltrami operator for 2D-manifold (surface meshes). I'm wondering if it is possible to define Laplace-Beltrami operator for 1D-manifold. If this is ...
0
votes
1answer
188 views

Locally Euclidean

A quick definition check: Would someone please tell me what "locally Euclidean" means when applied to a Riemannian metric? I have kind of an idea about it, for example when we multiply by a ...
6
votes
1answer
352 views

Closed and Exact forms/deRham groups

I'm trying to translate these theorems, below, into theorems about vector and scalar fields in $\mathbb R^n\setminus\{0\}$, in the case $n = 2$. First Theorem: Let $A = \mathbb R^n\setminus \{0\}$, ...
3
votes
0answers
82 views

Closed / embedded surface

Given a closed surface in $\mathbb R^3$, is it necessarily an "embedded surface"? I think it is true, but that is just because I can't think of a closed surface for which we cannot construct a smooth ...
2
votes
1answer
378 views

Is the union of two manifolds a manifold?

Suppose I have $M$ and $N$, two $k$-manifolds in $\mathbb{R}^n$. Is it true that $M\cup N$ is also a manifold? What is a sufficient condition for positive answer?
1
vote
1answer
67 views

to show $g$ attains maxima and minima

Let $A$ be a symmetric $n\times n$ real matrix and define $G:\mathbb{R}^n\rightarrow \mathbb{R}$ by $G(t)=\langle At,t\rangle$; let $g:S^{n-1}\rightarrow \mathbb{R}$ be the restriction of $G$ to the ...
2
votes
1answer
150 views

ellipticity of the Laplacian associated to the de Rham complex

I am struggeling with the following comment that I read regarding the de Rham complex: Define $(d + \delta)_e : C^\infty(\Lambda^e(T^*M)) + C^\infty(\Lambda^o(T^*M))$ where \begin{equation} ...
1
vote
0answers
89 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 ...
3
votes
1answer
167 views

The diffeomorphism of $\mathbb R^n$

If $f$ is a diffeomorphism of $\mathbb R^n$ and $K$ is a compact set in $\mathbb R^n$, can we find another diffeomorphism $\tilde f$ of $\mathbb R^n$ such that: (1)$f=\tilde f$ on a neighborhood of ...
11
votes
1answer
345 views

What is the universal property of the tangent bundle of a smooth manifold?

The process of writing my own notes on smooth manifolds have led me to wonder about this. All I've really found is the following: In addition to Madame Ehresmann's references, there is in ...
2
votes
0answers
599 views

Pulling back vector fields

I want to find conditions under which one can pull-back vector fields (if it is at all possible). Let $F:M \to N$ be a smooth surjective map between two $C^{\infty}$ manifolds of the same dimension. ...
0
votes
1answer
294 views

diffeomorphism of derivative map at tangent space level

$f: X\rightarrow Y$ is a diffeomorphism, then at each $x$ its derivative $df_x$ is an isomorphism of tangent spaces.could you please give me proof and insight of this result?
3
votes
2answers
377 views

divergence of a vector field on a manifold

I've been asked to show the following: For a vector field $V$ on a semi-Riemannian manifold with metric $g$ that $$Div \cdot V = \frac{1}{\sqrt{\det(g)}}\partial_i\left(\sqrt{\det(g)}V^i\right)$$ I ...
3
votes
3answers
271 views

Scalar product on manifold.

Let $M$ be a closed Riemannian manifold and $\omega$ and $\eta$ two differential forms of the same degree. Then one can consider $\int_M \omega \wedge *\eta$, where $*$ denotes the Hodge star ...
2
votes
1answer
99 views

Did I integrate a differential form correctly?

I start with 1-form $\omega=f\,dx$ on $\left[0,1\right]$ where $f\left(0\right)=f\left(1\right)$ and a $g:\left[0,1\right]\to R$ with $g\left(0\right)=g\left(1\right)$ and I want to integrate ...
4
votes
1answer
346 views

Stokes theorem for Lorentz manifolds

Reading Tao's book: Nonlinear Dispersive Equations I came upon an identity (the energy flux identity for the wave equation, page 90) for which the proof uses the Stokes theorem. In this case he uses ...
2
votes
1answer
115 views

Translating theorem on closed/exact forms

I'm trying to translate this theorem, below, into theorems about scalar and vector fields in $\mathbb R^3$: Theorem: Let $A$ be a star-convex open set in $\mathbb R^n$. Let $\omega$ be a closed ...
0
votes
0answers
120 views

Derivative/Chain Rule (for MANLYfolds) Computation

Embarrasingly, I can't compute the following derivative. $dh(X)=\left.\frac{d}{dt}h(e^{XT}]\right|_{t=0}$, where $X$ resides in the lie algebra of $\rm SL(3,\Bbb C)$ [ie $\mathfrak{sl}(3,\Bbb C)$] ...
2
votes
1answer
141 views

DeRham Cohomology

Let $p$ and $q$ be two points of $\mathbb{R}^n$ where let $n\geq 1$. Then $$\dim H^k(\mathbb R^n - p - q) = \begin{cases}0, &\text{ if }k\text{ is not equal to }n-1,\\ 2,&\text{ if }k = ...
1
vote
1answer
115 views

group of automorphism of a vector space

While reading manifold theory I stuck to this problem: $V$ be a vector space with $\dim V<\infty$ over $\mathbb{R}$ and $GL(V)$ be the group of all linear isomorphisms of $V$ into itself. A basis ...
1
vote
1answer
169 views
2
votes
2answers
148 views

Does a non-zero wedge product make a coordinate system?

I already know that a coordinate system must have a non-zero wedge product of the components, but does it go the other way, that is, does $df_1\wedge\cdots\wedge df_n(p)\ne0$ mean that $f$ is a ...
0
votes
1answer
78 views

Transforming vectors to coordinate vectors

Let $X_1,\dots,X_n$ be n vector fields on an open subset $U$ of a manifold of dimension $n$ Suppose that at $p\in U$, the vectors $(X_1)_p,\dots,(X_n)_p$ are linearly indipendent. would any one say ...
1
vote
1answer
99 views

Coordinate system with $df_1\wedge…df_n$

Is it possible to have a coordinate system $f:M\to R^n$ with $df_1\wedge...\wedge df_n(p)=0$?
1
vote
2answers
643 views

Question about Christoffel symbols

In my lecture notes I have two definitions of the Christoffel symbols. The first is the smooth functions $\Gamma^k_{ij}: U\subseteq M\to\mathbb{R}$ defined for $i,j,k=1,2$ by ...
2
votes
2answers
134 views

Differential Geometry Question

How to prove the following: Show that there is no metric on $S^{2}$ having curvature bounded above by $0$ and no metric on surface of genus $g$ which is bounded below by $0$.
1
vote
1answer
109 views

parametrisation of a curve

Is it always possible to choose a continuous parametrisation (only continuous) of a piecewise smooth curve to make it smooth? Assume we have this curve in $\mathbb{R}^{n}$.
1
vote
1answer
170 views

Wedge product $d(u\, dz)= \bar{\partial}u \wedge dz$.

How to show that if $u \in C_0^\infty(\mathbb{C})$ then $d(u\, dz)= \bar{\partial}u \wedge dz$. Obrigado.
2
votes
1answer
1k views

Calculating mean and Gaussian curvature

I am stuck on this question from a tutorial sheet I am going through. Compute the mean and Gaussian curvature of a surface in $\mathbb{R}^3$ that is given by $z=f(x)+g(y)$ for some good functions ...
3
votes
1answer
381 views

Covariant Derivatives and the Cross Product

I've recently read a paper that used a covariant derivative product rule for cross products. It was something like $\nabla_v (A \times B) = (\nabla_v A) \times B + A \times (\nabla_v B)$. Here, $A, ...
4
votes
0answers
95 views

The Nambu bracket

Does anybody know how to show the Jacobi identity for the Nambu bracket in $\mathbb{R}^3$? The Nambu bracket with respect to $c \in \mathcal{F}(\mathbb{R}^3)$ is defined as $$\{F,G\}_c = \langle\nabla ...
3
votes
0answers
316 views

Parallel transport in discrete differential geometry - programming a game

I would like to get a better intuitive grip on how parallel transport works. I once saw a video a German guy made with a little car having a gyroscope. That car was dragged on a big beach ball and the ...
2
votes
1answer
94 views

normal bundle of a boundary

let $X$ and $Y$ be compact, oriented manifolds and assume that $\partial X=Y$. Is it true that the normal bundle of $Y$ in $X$ is trivial? if it is the case, is there a simply explaination? Thanks
0
votes
1answer
140 views

Uniqueness of smooth structure on a zero-dimensional smooth manifold

In John Lee's Book "Introduction to Smooth Manifolds" on page 17, Example 1.12, the author states that the smooth structure on any zero - dimensional manifold is unique. That confuses me, suppose for ...
3
votes
2answers
260 views

Asymptotic Expansion for heat operator $e^{-t\triangle}$

I'm afraid the question below might turn out to be very stupid - I just don't know how to make sense of two asymptotic expansions, given the heat operator $e^{-t\triangle}$ with $\triangle$ a ...
2
votes
0answers
108 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 ...
0
votes
0answers
72 views

How to show this equation (hypersurface, differential geometry, calculus)

Suppopse $X:\mathbb{R}\times (0,T) \to \mathbb{R}^2$ is a parametrisation of a smooth curve $\Gamma(t)$ with $X(a + 1, t) = X(a, t)$ for all $a$. Let $v:\mathbb{R}\times (0,T) \to \mathbb{R}$ be a ...
0
votes
0answers
87 views

Elementary doubt regarding a tangent vector

The book I am using defines a tangent vector to $\mathbb R^3 $ at a point $p$; $\ v_p $ as the line segment $\ p+v $ though both p and v are points in $\mathbb R^3 $. My question is since all points ...
1
vote
0answers
137 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 ...
0
votes
1answer
160 views

Finding the point on a parametric curve/surface that lies normal to a known point off the surface

I'm given a periodic parametric curve $P = ( x(t),y(t) )$, where $t \in [0, 2\pi)$ and $P(2\pi)=P(0)$. I have a point $F = (x,y)$ that is not on that surface. Could someone tell me how to find the ...
3
votes
1answer
190 views

Embedding/Submersion Properties of Cotangent Maps (Pullbacks)

Let $M$ and $N$ be smooth manifolds and $f: M \to N$ a smooth map. Define the pullback bundle $\pi_f^*:f^*(T^*N) \to M$ as usual by $ f^*(T^*N) = \{(x,j^1_{f(x)}g) \in M \times T^*N \}$ with ...
5
votes
1answer
677 views

The winding number and index of curve

If $\gamma $ is a smooth closed curve in $\mathbb R^2-\{0\}$, I want to know whether the winding number of $\gamma $ about $0$, i.e., $\frac{1}{{2\pi i}}\int\limits_\gamma {\frac{1}{z}} dz$ is equal ...
2
votes
1answer
526 views

Parametric Equations for a $2$-torus

I know that for a torus (with one hole) the parametric equations describing it are $x= (c + a\cos v)\cos u, y= (c + a\cos v)\sin u, z= a\sin v$, where $c$ is the radius from the center of the hole to ...
2
votes
0answers
151 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 ...
4
votes
2answers
171 views

How to decide that a curve segment is not an ellipse line segment?

Let me ask a question , given any short curve segment , how can you decide that it is not an ellipse line segment by a finite calculations? Thank you in advance.
3
votes
1answer
473 views

The square root of positive definite matrix

Let $M$ be the manifold of real positive definite $n \times n$ matrices, define a mapping $i:A \to \sqrt A$ (where $A\in M$ and $\sqrt A$ means the unique positive definite square root of $A$). Please ...
0
votes
0answers
128 views

Given Poincare Polynomial find the manifold.

Suppose we have a polynomial, is it always the Poincare polynomial of some manifold? I guess the answer is no, but don't know any example. Even more, if we have a ring, is it the cohomology ring of ...