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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How to define a Holder seminorm of a section

I'm reading "Variational Problems in Geometry",Seiki Nishikawa, in the figure below. Let $(M,g)$ be a compact $m$ dimensional Riemannian manifold with no boundary. $T>0, 0<\alpha<1, ...
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144 views

Searching for a thesis.

In several articles about curves, etc, in Minkowski spaces $\Bbb L^3$ and $\Bbb L^4$, there is Walrave, J., Curves and surfaces in Minkowski space. Doctoral Thesis, K.U. Leuven, Fac. of ...
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42 views

Question about complete metric on manifolds

I've recently been wondering about whether non-complete metrics on manifolds can be transformed into complete metrics on manifolds and whether all manifolds have complete metrics. After some googling ...
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Computing volume element in spherical coordinates

Suppose $y = (r, \theta^1, \theta^2)$ are spherical coordinates in $(\mathbb{R}^3,g)$. What is the $d\text{vol}$ in these coordinates? I solved it but I don't know if it's right. My solution: We ...
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1answer
30 views

What is the metric on a cone?

I'm trying to learn differential geometry. I thought a cone would be an easy place to start with calculating a metric, shape operator, what have you. First of all, by the way, when I say "cone" I ...
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1answer
31 views

Flow of a Metric & Conformality

What is the flow of a metric mathematically? I want to be able to understand what it means to say that a metric preserves a conformal structure through the actual definition $\Theta_t^*g = ...
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20 views

Inverse of the Stereographic projection $\mathbb{CP}^1 \to S^2$

I've some problems with this exercise: Consider the stereographic projection $$ \varphi \colon S^2 \setminus \{ (0,0,1)\} \to \mathbb{CP}^1\setminus \{[1:0]\}$$ $$ (x,y,z) \mapsto [ x+iy:1-z]$$ and ...
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55 views

Degree of odd mapping of sphere

Is it possible to prove the fact that every smooth odd mapping of $S^n$(such that $f(x)=-f(-x)$ for every $x$) has odd degree using formula which connects degree and number of preimages of regular ...
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26 views

Ricci scalar algebra

Is this derivation correct? If $R=0$, $R$ being the scalar curvature, then: $$R_{;k}=0$$ $$(g^{ac}g^{bd}R_{abcd})_{;k}=0$$ $$g^{ac}g^{bd}(R_{abcd})_{;k}=0$$ $$R_{abcd;k}=0$$
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1answer
21 views

On maximality of differentiable structure

In the book Foundations of Differentiable Manifolds and Lie Groups (by Warner) the author defines a differentiable manifold to be a pair $(M, F)$ where $F$ is a maximal atlas. Usually in the ...
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1answer
65 views

Exponential map and distance on a Riemannian Manifold

I'm trying to solve an exercise which I thought at first seemed simple but I'm having some trouble pegging down the error term. The question is to prove that on a Riemannian manifold we have ...
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1answer
35 views

Conformal classes and almost-complex structures

It is well-known that on closed oriented surfaces $S$, conformal classes of metrics on $S$ correspond bijectively to complex structures on $S$. My understanding is that this correspondance goes as ...
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1answer
38 views

When are two submanifolds “the same”?

Consider two smooth submanifolds $N\subseteq\mathbb R^n$ and $M\subseteq\mathbb R^m$. Let there be a function $\varphi\colon N \to M$ that is bijective. Which properties does the function $\varphi$ ...
3
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3answers
104 views

Differentiability of the distance function

Suppose that $d:X \times X \to \mathbb{R}$ is a geodesic distance function on a smooth Riemannian manifold $X$ ($d$ is determined by metric tensor) and $x \in X$ is fixed. What can be said about the ...
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29 views

Unipotent representations of SL(2,R) by quantization

I'm a PhD student in mathematical physics and I happen to need some elements of Kirillov's "orbit method" for producing representations of Lie groups. I'm familiar with symplectic geometry, geometric ...
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1answer
45 views

Flow of a differential equation over what interval

Let $\dot{x}=x^2$. Over what interval is the flow defined? I can see that the solution is of the initial value problem $\dot{x}=x^2$, $x(0)=x_0\ $ is $$ x(t)=\frac{x_0}{1-x_0\cdot t}$$ and that it ...
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1answer
62 views

Gauss curvature of the graphs of the real and imaginary parts of an analytic function. [closed]

General question about analytic function f of complex variable z: $ f(x + i y) = f(z) $. At any mapped point, are Gauss curvatures in separate 3D plots of real and imaginary parts of $ f(x + i y)$ ...
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1answer
36 views

The isometry group of the simply-connected Ricci-flat closed manifold

If $M$ is a simply-connected Ricci-flat closed manifold, then is $I(M)$ the isometry group finite?
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59 views

does a commutative diagram implies pull-back?

Let $\xi=(E,p,B),\xi'=(E',p',B')$ be fibre bundles. Let $f: B\to B'$, $\bar f: E\to E'$ be maps such that the diagram commutes $\require{AMScd}$ \begin{CD} E @>\displaystyle\bar f>> E'\\ ...
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44 views

Exercise about tangent space

I have just started of learning about Manifolds from Milnor's book.I am stuck on following exercise,give me some idea. Let $U$ be an open subset of a manifold $X$. Show that for any $x \in U$ the ...
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1answer
66 views

What are some interesting topics for a differential geometry essay?

As an assignment I have been asked to write an essay on an aspect related to differential geometry. My lecturer has proposed topics such as architecture, navigation and computer graphics. I personally ...
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1answer
29 views

Jacobian of parametrized ellipsoid with respect to parametrized sphere

I'm not even sure how best to phrase this question, but here goes. Given $\theta$ (elevation) and $\phi$ (azimuth), the unit sphere can be parametrized as $ x = \cos(\theta)\sin(\phi) \\ y = ...
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1answer
26 views

Signature of a finite covering space

Suppose $\tilde{M}\rightarrow M$ is a finite (k-fold) covering of the smooth, oriented, compact 4-manifold $M$. Is there a relation between the signatures ...
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1answer
60 views

Distance between two points on the Clifford torus

How can I obtain the distance between two points $\mathbf{x}=(x_1,x_2,x_3,x_4)$ and $\mathbf{y}=(y_1,y_2,y_3,y_4)$ that belong to the $2$-torus $\mathbb{S}^1\times \mathbb{S}^1$? This is, I want to ...
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How do I determine sufficient conditions for the existence of the solution of an initial value problem?

Suppose that $f$ is a smooth function from $\mathbb R^{3}$ to $\mathbb R$ with $f(0,0,0)=0$. Under what sufficient condition will the differential equation $f(x,y,y')$ have a solution satisfying the ...
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1answer
74 views

Meaning of **Canonical metric** on complex manifolds

What is the meaning of Canonical metric on complex manifolds ?
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measuring the curvature of a pringle

Assuming for the sake of argument that a pringle shaped potato chip has a constant negative curvature, (which according to http://math.stackexchange.com/a/617610/88985 it is not ) see also picture ...
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1answer
32 views

The existence of a simply-connected neighborhood of a contractible loop

Let $M$ be a smooth manifold with a point $x_0$ on $M$ and a smooth loop $\gamma$ at $x_0$. If $\gamma=0$ in $\pi_1(M,x_0)$, then can we find a simply-connected open set $U$ around $x_0$ such that ...
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1answer
55 views

How can we show that $f_1f_2…f_k=0$ iff $\exists j$ st $f_j=0$?

Assume $V$ is an n dimensional vector space. $f_1,...f_k\in V^*,v_1,...,v_k\in V$ Define the symmetric k tensor $f_1f_2...f_k(v_1,..,v_k)=\Sigma_{\delta\in S_k}f_{\delta 1}(v_1)...f_{\delta_k}(v_k)$ ...
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1answer
24 views

Then, how can we show that $\forall i,j\in \mathbb Z, s.t.:1\leq i<j\leq n$, $e_i\wedge e_j $ is a basis vector for $\wedge^2(V) $?

Let $V$ be a n dimensional vector space. Suppose $x,y\in V, f,g\in V^*$. Define $f\wedge g(x,y) = det \left( {\begin{array}{cc} fx & fy \\ gx & gy \\ \end{array} } \right)$ Then, how can we ...
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Ricci SCALAR curvature

Are there any manifolds that have NULL SCALAR curvature but not null ricci curvature tensor? For dim>2, obviously. Edit: Are there, also, any manifolds of null scalar curvature but ricci curvature ...
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Principle Lines of curvature

In the text by Manfredo P. Do Carmo entitled Differential Geometry of Curves and Surfaces, an analysis of the principle directions is made near a non umbilic point on pp 160-161. I have followed his ...
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1answer
40 views

A question on self-dual differential 2-forms

This question is from Lemma 2 in Derdzinski's paper. Let $$\omega=e_1\wedge e_2+e_3\wedge e_4, \eta=e_1\wedge e_3+e_4\wedge e_2, \theta=e_1\wedge e_4+e_2\wedge e_3$$ be a basis for self-dual ...
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Rectifying linearly independent vector fields

Suppose we are given two vector fi elds $V_1$ and $V_2$ - defined on $R^n$- such that the vectors $V_1(x)$ and $V_2(x)$ are linearly independent for each $x$. Is it possible to find a diff ...
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23 views

question on tangent bundle

Let $X$ be a manifold and consider its tangent bundle $T(X)$ and let $p$ be the usual map $T(X) \to X$. Then why is it locally trivial ? i.e why for all $x\in X$ exist open neighborhood $U$ of $x$ ...
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The intuition of the rank theorem

In Rudin's Principle of Math Analysis, I know the proof of the rank theorem. However, I fail to understand its content. 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ ...
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1answer
23 views

How do I show that if there is a diffeomorphism between two smooth manifolds, the manifolds have the same dimension?

I think I've somehow got to bring tangent spaces into the picture, but how do I solve this?
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1answer
16 views

Coordinate dependence of the volume of parallelotope

It is well known that for $n$ vectors $v_1, \ldots, v_n$ in $\mathbb R^n$, the determinant of the matrix $A = (v_1 \ldots v_n)$ [i.e. with the vectors as columns] is related to the volume of the ...
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2answers
44 views

What does it mean to take the cross product of velocity and acceleration?

This is from a practice question I am working on. The osculating plane to the curve given by the vector valued function $r(t) =\langle\cos(t), (t-1)^2, -\sin(t)\rangle$ at the point corresponding ...
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1answer
27 views

Finding bump function on a smooth manifold using partitions of unity.

Let $M$ be a smooth manifold. Let $A$ and $B$ be disjoint closed sets of $M$. Show there exists a smooth function $f$ such that $f^{-1}(0)=A$ and $f^{-1}(1)=B$. This is my idea so far, Since $A$ ...
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0answers
30 views

How to derive the cigar soliton solution to the Ricci flow equation?

I am trying to derive the cigar soliton solution to the Ricci flow equation. Such solution has the form $$ {\frac {{{\it dx}}^{2}+{{\it dy}}^{2}}{{{\rm e}^{4\,t}}+{x}^{2}+{y}^{2 }}} $$ I am ...
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Exponential map on Diffeomorphism group of $S^1$

I am reading Segal book on Loop groups, and he mentions the following theorem: $$ \exp: Vect(S ^1) \rightarrow Diff(S ^1) $$ the map taking a vector field to the diffeomorphism obtained by flowing ...
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Why do we need an orientable surface for Gauss map?

I'm learning Differential Geometry recently with do Carmo's book. In the book, Gauss map is define as a differentiable map from an orientable surface $\mathcal{S}$ to $S^2$ in such a way that for ...
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31 views

Reparametrisation of closed not closed

I would like an example of a closed curve and a reparametrisation of the same curve that is not closed.
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21 views

$d+\Gamma$-understanding connections

$d+\Gamma$ is a formula commonly found in textbooks when one reads about connection. To be more precise: if $U$ is an open set over which a given vector bundle $E \to M$ is trivial then a coonection ...
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40 views

The curvature of surfaces in Euclidean space (Theorema Egregium)

The below animation is from Wikipedia. It shows how a helicoid can be deformed into a catanoid and vice versa without stretching. Because of this, the Theorema Egregium shows that the Gaussian ...
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Retract of a free $\Omega(\mathbb{R})$-module

Can an open subset X of $\Omega(\mathbb{R}^2)$ be an $\Omega(\mathbb{R})$-module retract of some free $\Omega(\mathbb{R})$-module? Here $\Omega(\mathbb{R}^n)$ denotes the usual topology of ...
2
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80 views

Pushforward along an exponential map

My differential geometry is a bit rusty, so I'd like some help with what follows: I have the following setting: $M,N$ Riemannian manifold of dimension $m<n$ with codimension $d$, $M$ in embedded ...
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$K = 0$ and $K = \text{const}$ surfaces produce $k_g = \text{constant}$ intersections?

Generally speaking, when do constant K (Gauss curvature) and zero K surfaces intersect to produce lines of constant geodesic curvature $ k_g $ ? Small circles on a sphere are examples. Or more ...
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49 views

What is the difference between abstract index notation and Ricci index notation?

I'm reading Straumann's GR text and he talks about the difference between abstract index notation and Ricci index notation very briefly. So I read the wiki article, but that did not help much. Say we ...