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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354 views

Circular Helicoid

A helicoid has the following parametric equation: $$ r(u,v)=\left \langle v\cos u,v\sin u,cu \right \rangle,\qquad u,v,c\in\mathbb{R}. $$ In ruled form, $$r(u,v)=\alpha(u)+v\Lambda(u),$$ it has ...
4
votes
1answer
151 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
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4answers
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*understanding* covariance vs. contravariance & raising / lowering

There are lots of articles, all over the place about the distinction between covariant vectors and contravariant vectors - after struggling through many of them, I think I'm starting to get the idea. ...
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vote
0answers
27 views

Is there a mathematical treatment of continuously (or smoothly) deforming surfaces, or, in general, continuously deforming manifolds?

I'm wondering if there's been any attempts to construct formal mathematical descriptions of deforming surfaces, whether evolving to some steady state configuration or reacting to some pulse, force, or ...
0
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1answer
24 views

Prove that this is a smooth surface

S is the surface satisfying $$f(x, y, z) = z^2 + (\sqrt{x^2+y^2}-a)^2 -r^2 =0$$ where $a,r\in\mathbb{R}$ Prove that $S$ is a smooth surface. Do we differentiate with respect to $x, y$ and $z$ ...
4
votes
1answer
171 views

Gaussian curvature and mean curvature sufficient to characterize a surface?

Is the knowledge of Gaussian and mean curvature (and thus of the principal curvatures) sufficient to characterize a surface uniquely? If not, is there another geometric quantity one can add to obtain ...
2
votes
2answers
404 views

Gentle introduction to quasi-geodesics

Compared to the concept of geodesics the concept of quasi-geodesics seems to be substantially harder to grasp and digest. I was given a promising hint to the concept of quasi-geodesics here but the ...
5
votes
1answer
2k views

Riemann, Ricci curvature tensor and Ricci scalar of the n dimensional sphere

I am calculating the Riemann curvature tensor, Ricci curvature tensor, and Ricci scalar of the n sphere $x_0^2 + x_1^2 + ....+x_n^2=R^2$, whose metric is $$ds^2=R^2(d\phi_1^2 + \sin{\phi_1}^2 ...
4
votes
1answer
162 views

Zeros of the second fundamental form

Let $ f:M \rightarrow N $ be a minimal immersion (of arbitrary codimension or an hypersurface if it is necessary) and let $ |A| $ be the norm of its second fundametal form.If $ A $ is not identically ...
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0answers
42 views

Computing relative de Rham cohomology?

Let $f: N \to M$ be a smooth map. Then in Bott and Tu they define a relative de Rham complex $\Omega^{\bullet}(f)$ where $\Omega^k(f) = \Omega^k(M) \oplus \Omega^{k-1}(N)$ with coboundary map given in ...
3
votes
1answer
36 views

Ricci flow on compact surfaces flows the metric conformally

The (normalized) Ricci flow on compact surfaces is given by $$\frac{\partial}{\partial t}g_{ij}=(r-R)g_{ij}\text{ ,}$$ and in the beginning of Hamilton's paper on the topic he points out that since ...
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0answers
25 views

References for actions of infinite-dimensional Banach-Lie groups on infinite-dimensional Banach manifolds

I am starting to study infinite-dimensional manifolds, specifically, Banach manifolds. I found some interesting introductory texts in which the mathematical background is developed with some detail. ...
2
votes
1answer
394 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
0
votes
1answer
20 views

Why the well-defined of Gauss map depends on surface is orientable?

Let $S$ is a surface. Define a mapping $g:S\rightarrow S^2\subset R^3$ of $S$ into the unit sphere $S^2$ , associating to every $p\in S$ a unit vector $N(p)\in S^2$ normal to $T_pS$. Why the ...
0
votes
1answer
29 views

assumptions for existence of envelope of a family of curves

Given a family of curves $F(x, y, c) = 0$ for $c$ in a range of real numbers, the envelope $E$ is the curve tangent to every member of the family. I see that it is defined by the solution of $F(x, ...
1
vote
1answer
43 views

Linear algebra for modern differential geometry( and other types of modern geometry, like analytic, complex and algebraic)

I wish to study real and complex analysis(for example, Pugh "Real Mathematical Analysis" and Cartan "Elementary theory of analytic functions of one and several complex variables") and modern ...
1
vote
1answer
103 views

Parabolic Cusp of an Action on the Upper Half Plane

This is a basic definition question. Parabolic bundles are used in certain counting arguments in my research area. I asked my advisor for a reference on these, and he directed me to the paper of Mehta ...
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votes
0answers
17 views

How to reparametrize a geodesic such that $\nabla_V V = 0$

In Nakahara's section on geodesics he says that $\nabla_V V = fV$ can be reparametrized to give $\nabla_V V = 0$ if we use the reparametrize the tangent vector components $t \rightarrow t'$ such ...
5
votes
1answer
152 views

Quotients of $S^{2n+1}$

Any sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$ Now say one wants to quotient it ...
6
votes
1answer
224 views

Geometric Interpretation: Parallel forms are harmonic

Let $(M,g)$ be a Riemannian manifold. The canonical volume form $\mu=\sqrt{\det g_{ij}}\mathrm{d}x^1\wedge\dots\wedge\mathrm{d}x^m$ is parallel w.r.t. the induced Levi-Civita conection $\nabla$ ...
2
votes
1answer
292 views

“Completing” a vector field on a non-compact manifold $M$

Suppose I have a non-compact smooth manifold $M$ and an arbitrary nowhere-vanishing smooth vector field $X$ on $M$ which is not complete. Is there a way to create a smooth vector field $V$ that is ...
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votes
2answers
92 views

Show that , for all $(s_{0},t_{0})\in [0,1]\times [0,a]$, the curves $s\to f(s,t_{0})$, $t\to f(s_{0},t)$ are orthogonals.

Let $f:[0,1]\times [0,a]\to M$ a parameterized surface such that for all $t_{0}\in[0,a]$, the curve $s\to f(s,t_{0})$, $s\in [0,1]$, is a parameterized geodesic by arc lenght , orthogonal to the ...
0
votes
1answer
15 views

Show $2xx' + 2yy' + 2zz' = 0$ for curve on sphere.

This is from Manfredo P. do Carmo's Differential Geometry of Curves and Surfaces. I have just been introduced to orientations of regular surfaces, the Gauss map and the differential of the Gauss map, ...
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3answers
196 views

What is the relation between dx in elementary calculus and dx in differential geometry?

I've recently started studying differential geometry and was really hoping that in doing so I'd finally have an answer to something that's been bugging me since I first learnt calculus - what is ...
16
votes
3answers
598 views

How to prove Lagrange multiplier theorem in a rigorous but intuitive way?

Following some text books, the Lagrange multiplier theorem can be described as follows. Let $U \subset \mathbb{R}^n$ be an open set and let $f:U\rightarrow \mathbb{R}, g:U\rightarrow \mathbb{R}$ be ...
5
votes
1answer
203 views

characterizing semi-Riemannian spaces of constant curvature

How does one characterize $n$-dimensional semi-Riemannian spaces of constant curvature? By "characterize," I mean giving both a definition and some insight into how the possibilities work out in ...
4
votes
1answer
129 views

Symplectic Chart

I was reading the article "Symplectic structures on Banach manifolds" by Alan Weinstein. In this article there is one theorem, which is as following: If $B$ is a zero neighborhood in Banach space. ...
1
vote
1answer
79 views

a special extension of a two variable function

We consider the function $f(x,y)=x^2+y^2$ in $\omega = (0,1)^2.$ I am wondering about the existence of a $C^2-$extension $F$ of $f$ in $\Omega = (0,2)^2$ such that $F$ is harmonic in ...
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vote
0answers
36 views

How to show $\nabla_a\nabla_c R_{ad}-\nabla_c \nabla _a R_{ad}=-R_{cade} R_{ae}-R_{ce} R _{de}$?

$\nabla$ is Riemann connection and $R_{ij}=g^{kl}R_{ikjl}$. How to show $\nabla_a\nabla_c R_{ad}-\nabla_c \nabla _a R_{ad}=-R_{cade} R_{ae}-R_{ce} R _{de}$ ? Or generate commutator of generate ...
0
votes
2answers
39 views

Two geodesics cannot form a simple region

Suppose S is an orientable surface with nonpositive Gaussian curvature. How can I prove that two geodesics that start from the same point $p\in S$ cannot meet again at another point $q\in S$ such that ...
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vote
0answers
18 views

Connected components of the complement of a geodesic

I came across the book "Knots, Molecules, and the Universe: An Introduction to Topology" by E. Flapan which is quite nice. In the first chapters it is discussed how one can distinguish the sphere ...
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vote
0answers
12 views

A piecewise regular simple closed curve bisects the area of the unit sphere if and only if it has total geodesic curvature 0

How can I prove that "A piecewise regular simple closed curve bisects (this curve splits the unit sphere into two pieces, the area of which are equal) the area of the unit sphere if and only if it has ...
2
votes
1answer
51 views

Is the cuspidal cubic $\{y^2 = x^3\} \subset \Bbb R^2$ not smooth?

Cuspidal cubic $y^2=x^3$ in $\Bbb R^2$ "seems to be not smooth" intuitively because its pictured graph has a cusp at the origin. But I read from book that it is a smooth manifold. I feel so confused. ...
0
votes
1answer
24 views

Definition of a smooth function between surfaces

If $S_1, S_2 \subset \Bbb R^3$ are two smooth surfaces, then what is the formal definition of a smooth map from $S_1$ to $S_2$? I am studying from Pressley's EDG, and the definition is given only in ...
0
votes
0answers
12 views

explaination of the metric tensor on another manifold?

In skew -product decomposition the following features are observed :- 1.the Riemannian Manifold $(M,g)$ has a product form of $$M=R\times \Theta$$ Where $\Theta ,R $ are connected $C^\infty$ ...
2
votes
2answers
44 views

Canonical notion of parallel transport

I have a "What is the right search term?" style question: Suppose $S\subset\mathbb{R}^3$ is a surface and that we are given two points $x,y\in S$. Furthermore, take $v_x\in T_x S$ to be a tangent ...
3
votes
1answer
46 views

Correspondence between vector bundles and locally free sheaves

I am trying to look at smooth manifolds in the context of locally ringed spaces. Vector bundles have a characterization in terms of sheaves of modules: if $X, \mathcal{O}_X$ is a topological (or ...
5
votes
3answers
420 views

Every manifold admits a vector field with only finitely many zeros

Let $M$ be a smooth manifold. I am trying to prove that $M$ admits a vector field with only finitely many zeros. This will follow if we can find a function $f : M\rightarrow \mathbb R$ such that ...
0
votes
1answer
19 views

Codifferential and Hodge star

Is this true, \begin{align} \notag \delta (f * \Omega )= f \delta (*\Omega)? \end{align} $\delta$ denotes codifferential, f is a function, $\Omega$ is a k-form and * is a Hodge star operator.
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0answers
16 views

“Reparameterization principle” and the way it is used

I am reading from Pressley's Elementary Differential Geometry. On page $79$, the author states a principle and says that it will be used throughout the rest of the book. Statement: The principle ...
4
votes
1answer
524 views

How does the boundary property usually work in PDE?

This may be related to the question Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?. In almost every chapter of a PDE textbook(e.g. Folland's Introduction to Partial Differential ...
4
votes
2answers
41 views

Is $f:SO(n)\rightarrow S^{n-1}$, $f(A)=(A^n_i)_i$ a submersion?

Let $f:SO(n)\rightarrow S^{n-1}$, $f(A)=(A^n_i)_i$, that is $f(A)$ is the last row of $A$. Show that $f$ is a submersion. I'm not sure how to calculate $df$, because I only know how to calculate the ...
3
votes
0answers
34 views

any open set in $\mathbb{R}^n$ is a $n$ dimensional manifold

I am trying to show this using the definition: M is a k-dimsensional submanifold of $\mathbb{R^n}$ if for all $x \in M$ the following condition holds: There exists an open set $U \subset ...
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vote
0answers
21 views

Gaussian curvature in polar coordinates

Find the expression for the Gauss curvature in the polar coordinates associated to the exponential map. I thought about using Gauss's lemma: if $(r,\theta)$ are polar coordinates in the tangent plane ...
2
votes
0answers
26 views

geodesic on an ellipsoid

Find all the geodesics which pass through the point $(a, 0, 0)$ on the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. What parametrization can i use to get the first fundamental form? ...
0
votes
0answers
19 views

Computing of proof of Li-Yau estimate

I try to compute the red line in picture below: \begin{align} \Delta(\partial_tL) +R\Delta L +\partial_t R &=\Delta(Q+|\nabla L|^2)+R(\frac{\Delta R}{R}-\frac{|\nabla R|^2}{R^2}) + \Delta R +R^2 ...
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votes
2answers
840 views

Calculating the Lie algebra of $SO(2,1)$

I am trying to calculate the Lie algebra of the group $SO(2,1)$, realized as $$SO(2,1)=\{X\in \operatorname{Mat}_3(\mathbb{R}) \,|\, X^t\eta X=\eta, \det(X)=1\},$$ where $$\eta = \left ( ...
0
votes
1answer
24 views

finding an integral manifold of a distribution

I have vector fields $\begin{cases}X_1 &=& -y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\\ Y &=& (x-y)z\frac{\partial}{\partial x} + (x+y)z \frac{\partial}{\partial y} + ...
2
votes
2answers
64 views

What does $C^k$ at a single point mean?

I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean? Furthermore if $f$ is $C^k$ at $p$, is it ...
0
votes
1answer
17 views

Chain rule for general manifolds

So, I need an explanation why shall it be $\frac{d}{dt}|_{t=0}(f(\phi_X^t(m)) = d_m f \frac{d}{dt}\phi_X^t(m)|_{t=0}$ where $\phi_X^t$ is a flow, m is point in a given differentiable manifold M. I ...