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)

4
votes
1answer
182 views

injective map in cohomology theory

I have the following question, which I dont really know if its true: Let $g : X \rightarrow Y$ be a continous map between two closed, oriented $n-$dimensional manifolds such that $g^{*} : H^{n}(Y, ...
3
votes
1answer
95 views

Linear independence regarding Exterior Power .

I have been trying to learn the proof of dimension of exterior power from this text : http://www.thehcmr.org/issue1_2/poincare_lemma.pdf.( Page 16) I am not able to understand the part of linear ...
0
votes
1answer
680 views

What does it mean to say a boundary is $C^k$?

I need a explanation on what does it mean to say a boundary is $C^k$. Can anyone help me please. And also need some explanation on how to straighten boundary ?
5
votes
1answer
301 views

differential form

one form $\alpha$ over a smooth manifold is non vanishing means for every $p\in M$, $\alpha_p\neq 0$. But $\alpha_p$ is linear map $T_M\to \mathbb R$, hence $\alpha_p(0)=0$. So confusion arises ...
6
votes
1answer
282 views

notation of derivation in differential geometry

I can't wrap my head around notation in differential geometry especially the abundant versions of derivation. Peter Petersen: Riemannian Geometry defines a lot of notation to be equal but I don't ...
5
votes
2answers
953 views

Proof that angle-preserving map is conformal

Let $\phi: S \to \bar{S}$ be a diffeomorphism between two surfaces in $\mathbb{R^3}$. Such a map is called conformal if for all $p \in S$, and $v_1, v_2 \in T_p(S)$ (the tangent plane) we have ...
1
vote
1answer
162 views

Tensored vectorspaces isomorphic to the endomorphisms [duplicate]

Possible Duplicate: Understanding isomorphic equivalences of tensor product I have the following question: Let $V$ be a vectorspace with an inner product $<.,.>$. Let $V^{*}$ be its ...
6
votes
2answers
306 views

Explicit computation of the Hodge codifferential

Question I'm given a Laplacian $\Delta_n=-4y^2 \cdot \frac{\partial^2}{\partial\bar{z} \partial z} + 4 iny \cdot \frac{\partial}{\partial\bar{z}}$, and I want it to be the Laplace operator associated ...
1
vote
1answer
51 views

Behavior of $L^2$-spaces under conformal variation

Consider a Riemannian manifold $(M,g_0)$ which is the interior of a compact manifold $(\overline{M}, \overline{g})$. I'm interested in a kind of conformal variation of the background metric $g_0$. ...
0
votes
2answers
186 views

positive non-constant harmonic function $f $ in $L^1(M)$ on a complete Riemannian manifold

Let $M$ be a complete Riemannian manifold, does there exists a positive non-constant harmonic function $f \in L^1(M)$? Who can answer me or give me a counter example? Thank you very much!
1
vote
1answer
151 views

The extension of diffeomorphism

Let ${\Omega _1}$,${\Omega _2}$ be two open sets in $\mathbb R^n$ and $f$ is a diffeomorphism between them. For every $x$ in ${\Omega _1}$, is there an open set $\Omega_{x} \subset \Omega_1$ and a ...
1
vote
1answer
190 views

Dimension of the space of matrices with constant determinant.

I'm looking for the dimension of the space of $n\times n$ real matrices $A$ such that $\det(A)=c$. I apply 2 different approaches and I get different answers. which one is correct? 1) So we ...
3
votes
1answer
442 views

Sufficient Conditions for Ricci Tensor to be Diagonal

What are the strongest (or most useful) conditions on a metric for it's Ricci tensor to be diagonal? I've read that if the metric is explicitly dependent on only one variable then the Ricci Tensor is ...
3
votes
2answers
104 views

Proving that a particular submanifold of the cotangent space is Lagrangian

I have the following problem in my differential topology class: Let $M$ be an $n$-dimensional manifold and let $\omega$ denote the standard symplectic form on the cotangent space $T^*M$. Let $f \in ...
1
vote
1answer
68 views

Variations in a Riemannian Manifold

Let be $M$ a Riemannian manifold and $X,Y$ vector fields over $M.$ Now take $p\in M$ arbitrarily, my question is, how construc a variation $f:U\to M,$ $$U\subset ...
2
votes
1answer
141 views

Does the value of the covariant derivative at a point of the metric tensor depend only on the involved tangent vectors?

Let $\nabla$ be an affine connection on a pseudo-Riemannian manifold $(M,g)$. Let $c:[0,1] \rightarrow M$ be a differentiable curve and consider vector fields $Y,Z$ along $c$. Is it true that the ...
1
vote
0answers
82 views

$C_0$ Convergence of Metrics under Ricci Flow when $\chi(M) = 0$.

I am a beginning graduate student, and I am trying to learn basic aspects of Ricci Flow via the uniformization theorem for compact surfaces. I am reading Chow and Knopf's introductory book as well as ...
2
votes
2answers
116 views

What is the limit distance to the base function if offset curve is a function too?

I asked a question about parallel functions in here . I understood that offset curves that are the parallels of a function may not be functions after J.M.'s answer. I got new questions after that ...
1
vote
0answers
63 views

Set of points where an application has rank $m$ is a smooth manifold.

Can someone help me with this problem? I have a $C^1$ function $G\colon\mathbb{R}^n\rightarrow \mathbb{R}^m$, where $k=n-m> 0$. If $M$ is the set of points $x\in G^{-1}(0)$ such that $(DG)_x$ has ...
9
votes
2answers
467 views

Variety vs. Manifold

In the ambit of differential geometry the aim is to study smooth manifolds. Why the objects studied in algebraic geometry are called algebraic varieties and not for example algebraic manifolds? I am ...
3
votes
2answers
890 views

How to find the tangent space to a matrix space

I have a hard time approaching these types of problems. In an article it had claimed that the tangent space to all symmetric matrices with the same signature as $M$ at a matrix $M$ is the set of all ...
1
vote
0answers
158 views

The Closed disc $D$ is a manifold with boundary

It is obvious geometrically, but how one proves with a few words, analytically, the statement above?Additionally, if one has a smaller open disc $D_\epsilon$ of radius $\epsilon$ centered at a point ...
1
vote
1answer
466 views

Projective Space orientation

I'm trying to prove that the projective plane $\mathbb{P}^n$ is orientable is and only if $n$ is odd. To do that that, I have a hint,to prove that the antipodal map is orientation preserving if only ...
3
votes
0answers
145 views

Complete a set of functions to obtain a system.

Let $M^m$ be a $m-$manifold and $C=\{y^1,\dots,y^k:U\subset M\rightarrow \mathbb{R}\},\;k<m,$ a collection of functions such that on a point $p$, we have that $dy^1|_p,\dots,dy^k|_p$ is linearly ...
3
votes
1answer
261 views

Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?

Let $G$ be a (EDIT: connected) Lie group of dimension $n$, and let $\mathfrak{g}$ be the associated Lie algebra. If $x_1,\ldots,x_n$ is a basis for $\mathfrak{g},$ is it necessarily true that the ...
1
vote
1answer
85 views

trouble with understanding notation - partition of unity, section

I am currently working with a book ("Fourier Integral Operators" by J.J. Duistermaat) that mentions a differential geometric construction that I struggle to understand. Here is the setting: Suppose ...
6
votes
0answers
297 views

Characterization of gradient vector fields

Let $V$ be a vector field on a smooth manifold $M$. Are there nice conditions under which there exists a (Riemannian) metric on $M$ such that $V$ is the gradient of some smooth function on $M$? One ...
2
votes
1answer
184 views

A question about orientation on a Manifold

Let $(U,x_1,x_2,\ldots , x_n)$ be a chart for a orientable manifold $M$, why $(U,-x_1,x_2,\ldots , x_n)$ is a chart for the Manifold $-M$, the same manifold with reversed orientation?
0
votes
1answer
110 views

Curves on a circle

Is it possible at every point $p=(x,y)$ on the unit circle, there is a continuous curve $C_p$ passing through it, a curve which is not only the single point $p$, and all these curves are pairwise ...
2
votes
1answer
130 views

Avoiding rationals $\implies$ constant

If $f:(0,1)\rightarrow\mathbb{R}^n$, with $n>1$, is a continuous curve in $\mathbb{R}^n$, with $f(p)=(x_1(p),x_2(p),...,x_n(p))$. Must the set of $p$ such that for some $i$, $x_i(p)$ is a rational ...
2
votes
0answers
229 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 $x = x_{0}t + \sqrt{1-t^2}y$, ...
6
votes
4answers
4k views

Simple proof of integration in polar coordinates?

In every example I saw of integration in polar coordinates the Jacobian determinant is used, not that i have a problem with the Jacobian, but I wondered if there's a simpler way to show this which ...
4
votes
2answers
279 views

The cone is not immersed in $\mathbb{R}^3$

The problem is to show that the cone $ z^2= x^2+y^2$ is not an immersed smooth manifold in $\mathbb{R}^3$.
1
vote
1answer
156 views

Curves and first fundamental form

Would I be right to think that if I have a coordinate system $(x,y)$ so that the lines/curves where one coordinate is fixed, so something like $x=a$ and $y=b$, always intersect at the same angle, then ...
1
vote
0answers
130 views

Why topological stratification is useful?

My main focus is on the applications of stratification in complex/abstract Algebraic geometry especially, from the scheme-theoretic viewpoint and (Added) Moduli spaces. I have a vague feeling that ...
3
votes
1answer
161 views

Coordinate change for metrics

I am rather confused by the idea of "geodesic polar coordinates", so I hope someone would kindly explain it to me. As far as my understanding goes, given a Riemannian metric ...
2
votes
3answers
293 views

Extending geodesics to vector fields

Let $c$ be a geodesic on a Manifold $M$. Some books define $c$ to be a Geodesic iff $\nabla_{c'}c'=0$. Therefore for every $c(t)$ the Geodesic must be extendable into a smooth vector field on an open ...
3
votes
0answers
252 views

moving a basis along a curve (parallel transport)

I'm considering a Riemannian Manifold $M^m$ and a Basis $\{X_1 ,...,X_n\}$ of the tangent space $T_pM$. When I consider now the parallel transport $E_i$ of the vectors $X_i$ along a curve c, then the ...
1
vote
0answers
58 views

Relations between metric on H and the disc model

In the book "Modular Forms" by Miyake one finds the definition of some obscure 'thing'. He calls it a metric on $\mathbb{H} = \{z \in \mathbb{C} : \operatorname{Im}(z) > 0 \}$. The following is ...
5
votes
1answer
139 views

$C^1$ approximation of a continuous curve.

Suppose I have two points $\alpha,\beta \in \Bbb{R}^n$. Define $$ X=\{\gamma \in C^1([0,1] , \Bbb{R}^n),\ \gamma(0)=\alpha,\gamma(1)=\beta ,0 <|\gamma'|<K\}$$ parametrized curves joining ...
3
votes
1answer
112 views

Could someone please explain what this question is asking?

I have some trouble understanding the following question: Suppose we have 1st fundamental form $E \, dx^2+2F \, dx \, dy+G \, dy^2$ and we are given that for any $u,v$, the curve given by $x=u, y=v$ ...
13
votes
2answers
462 views

Is the set of singular matrices ever a differentiable manifold?

I can see that invertible matrices are a differentiable manifold however I don't know how to show that something is not a differentiable manifold so easily. Is it ever the case that singular matrices ...
6
votes
1answer
417 views

Covector field on the sphere $S^2$ vanishing?

Covector field on the sphere $S^2$ vanishing? There exists a smooth vector field $X$ on $S^2$ that vanishes at exactly one point, for example at the north pole. My idea is the following: Let ...
4
votes
2answers
143 views

Change of variables to a desirable form

Suppose we have smooth function $\varphi: \mathbb{R}^n \to \mathbb{R}$ and let $x_0 \in \mathbb{R}^n$ be a non-degenerate critical point. That is, $$\begin{equation} \varphi'(x_0) = \nabla ...
1
vote
0answers
130 views

Energy displacement of a cylinder is at most $\pi r^2$.

I want to show that the energy displacement of $Z^{2n}(r)$ is at most $\pi r^2$. In the textbook of Mcduff and Salamon they write that I should identify the two dimensional ball (with radius $r$) ...
4
votes
0answers
194 views

Bi-invariant form on compact connected Lie group

Let $G$ be a Lie group and $\omega$ be a left invariant $k$-form, how to prove that $r^*_a \omega$ is left invariant? What I do: $(l^*_g (r^*_a \omega))_x (v_1, \ldots,v_k)=(r^*_a \omega))_{gx} ...
9
votes
3answers
978 views

Topology of the tangent bundle of a smooth manifold

I am having trouble to understand what topology is given to the tangent bundle of a smooth manifold that allows it to be a smooth manifold itself. In my understanding, among other things the topology ...
2
votes
1answer
409 views

are non-degenerate critical points always isolated?

I have a question regarding the isolation of critical points of a function: Suppose $f : \mathbb{R}^n \to \mathbb{R}$ is a $C^\infty$ function such that $f$ has a non - degenerate critical point at ...
3
votes
0answers
420 views

Need help understanding a lift of a vector field

This is a question from my differential geometry assignment: Let $\pi:M\to N$ be a submersion between two smooth manifolds and $X\in \Gamma(TN)$ is a vector field. We need to show that there ...
2
votes
0answers
179 views

Using Rayleigh Quotient to approximate the first eigenvalue of the Laplace operator on the unit disk

Let $D\subset\mathbb{R}^{2}$ unit disk, the first eigenvalue of the Laplace operator holds: $\lambda_{1}=\inf\left\{ \frac{\int_{D}\left|\triangledown ...