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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498 views

Generalized Laws of Cosines and Sines

I wonder the "laws of sines and cosines" in the two cases below and how to derive them. (or any related sources) (i) For geodesic triangles on a sphere of radius $R>0$. (so constant curvature ...
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178 views

Riemannian Connection (Very basic question)

We know that a connection $\nabla$ in a manifold M hashas the purpose of performing the same role as the covariant derivative of vector fields of surfaces in $\mathbb{R}^3$. Such analogies are ...
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2answers
141 views

help on connections

In the book on Riemannian Geometry by John Lee ("Riemannian manifolds: an Introduction to Curvature") the author gives an exercise on page 54 involving connections: Let $\triangledown$ be a linear ...
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824 views

Explicitly proving invariance of curvatures under isometry

I would like to know how to explicitly prove that Riemann Curvature,Ricci Curvature, Sectional Curvature and Scalar Curvature are left invariant under an isometry. I can't see this explained in most ...
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1answer
287 views

Need references on Cartan's method of moving frames.

Could anyone suggest a book or a paper containing a good, modern treatment to the Cartan's method of moving frames. Especially, I am interested in its use in studying geometric properties of surfaces ...
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1answer
302 views

The behavior of all unit speed geodesics on a surface of revolution.

In the $xz$-plane of $\mathbb R^3$, consider the closed non-singular curve $\gamma$ which is the image of the function $$t\mapsto (1+2\sin^2(t))(\cos(t),0,\sin(t)).$$ (Note that $\gamma$ is invariant ...
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239 views

Is the crossing number of a non-planar graph a function of the surface in which it is embeddable?

I know that a non-planar graph with one crossing can be embedded in a torus, and I expected that a graph with two crossings would require a double torus. This does not seem to be the case (cf. the ...
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1answer
281 views

How to show that the geodesics of a metric are the solutions to a second-order differential equation?

On $\mathbb R^n$, let $\rho: \mathbb R^n\to\mathbb R$ be a smooth function, and $g$ be the metric given by scaling the usual flat metric by $e^{2\rho}$. I want to know how to show that the geodesics ...
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221 views

Proof of Gauss's lemma in Riemannian geometry

In the proof of Gauss's lemma here, there is a step $\displaystyle\lim_{t\to0}\frac{\partial f}{\partial s}(0,t)=\lim_{t\to0}T_{tv}\ \exp_p(tw_N)=0$ However, the limit seems meaningless (unless ...
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2answers
1k views

A simple explanation of differential calculus and its link to geometry?

The wikipedia articles on differential calculus and differential geometry are quite long and not so straightforward for a layman like me. Is there a master of math vulgarization out there that could ...
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0answers
139 views

Differential map between smooth manifolds is smooth

Given a smooth map $f:M\to N$ between smooth manifolds how do you show that the differential map $df:TM\to TN$ is smooth?
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1answer
310 views

Projection map from TM to M is smooth

Given smooth manifold $M$ how do you prove that the projection map $\pi : TM\to M$, $(p,v)\mapsto p$ is smooth?
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161 views

Reference for densities and pseudoforms and non-tensorial representations of $\operatorname{GL}(n)$ and associated vector bundles

I'm looking for a reference that will set me straight on a few things. It started out with densities. In John Lee's book, "Introduction to Smooth Manifolds", densities on vector spaces are functions ...
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1answer
92 views

Submanifold of $\mathbb{R}^{n+m}$

I'm currently proving that three different definitions of a submanifold of $\mathbb{R}^{n+m}$ are equivalent, and I've mostly done it, but there's one implication that I'm struggling with. For $M ...
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1answer
94 views

What is the topological properties of this set

Let $x_1,\dots,x_n$ be $n$ points forming a rigid body in $\mathbb{R}^3$. The distance between each pair of points is constant. Let $R\in SO(3)$ be a rotation and $T\in\mathbb{R}^3$ be a translation. ...
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174 views

The diffeomorphism of neighbourhood

$M,N$ are two smooth $n$-manifolds and $A$ is a subset of $M$. A smooth mapping $f$ from $M$ to $N$ has the property that $df$ is nonsingular and $f$ is injective on $A$. Is there a neighbourhood $U$ ...
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1answer
69 views

Surface described by the graph and inequality of forms

Suppose that $S,T$ are two surfaces locally described by $(u,v,f_{1}(u,v))$ and $(u,v,f_{2}(u,v))$ where $f_{i}$ are maps from the tangent spaces of $S$ to $\mathbb{R}$ (respectively from T). (here ...
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0answers
77 views

Isoperimetric inequalities with relative perimeter

It is a well known result that if $\Omega\subset \Bbb{R}^N$ is an open set, with regular boundary (smooth, or Lipschitz) then the problem $$ \min_{E \subset \Omega |E|=c} Per(E)$$ has a solution, ...
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347 views

Higher-order derivatives in manifolds

If $E, F$ are real finite dimensional vector spaces and $\mu\colon E \to F$, we can speak of a (total) derivative of $\mu$ in Fréchet sense: $D\mu$, if it exists, is the unique mapping from $E$ to ...
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1answer
128 views

Show that the hyperboloid is a Riemannian manifold

Let G be the Minkovski quadratic form on $\mathbf{R}^{n+1}$ : $G(x,x)=-x_{0}^{2}+x_{1}^{2}+ \cdots +x_{n}^{2}$. Consider the (half) hyperboloid in $\mathbf{R}^{n+1}$ : $H=\{ x \in \mathbf{R}^{n+1} : ...
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1answer
83 views

Question on moment maps.

I have some trouble in understanding the notion of a moment map for the Lie group $S^1$: \ In the book "Moment maps, cobordism and Hamiltonian group actions" it is said on page 15, which you can find ...
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77 views

Is this function on the surface smooth?

Consider the following functions $F_{ij}:S\subset{\mathbb R}^3\to{\mathbb R}$, $$ F_{ij}(y) = \begin{cases} \frac{(y_i-x_i)(y_j-x_j)(y-x)\cdot n(y)}{|y-x|^3},&y\neq x; \\ 0,& ...
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1answer
168 views

Conceptualization of exterior powers of projective modules

Let $A$ be a commutative noetherian ring, and $P$ a projective $A$ module with $rank(P)=n$. I know that $\wedge^nP \simeq L$ for some rank 1 projective $A$-module, $L$; but I'm not sure of how to ...
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2answers
115 views

Does this updated $\frac{0}{0}$ two-dimensional limit exist?

This is a question closely related to the one I posted two days ago. Thanks to Christian Blatter's answer to that question, the limit (there are 9 limits here indeed.) $$ ...
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117 views

Which of these maps are alternating tensors?

I'm trying to decide which of the following are alternating tensors in $\mathbb R^4$ and express those that are in terms of the elementary tensors on R^4: $f(x,y) = x_1y_2 = x_2y_1 + x_1y_1$ ...
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2answers
708 views

Motivation behind the definition of a Manifold.

A manifold $M$ of dimension n is a topological space with the following properties: a) $M$ is Hausdorff b)$M$ is locally Euclidean of dimension n c) $M$ has a countable basis of open sets. Why is ...
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1answer
315 views

Prove that Killing vector fields form Lie algebra.

I want to find the integral curves of $[X,Y]$, then maybe can use this to prove. Can anyone gives an answer ? Thanks.
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0answers
183 views

$SU(2)$ is a covering space of $SO(3)$.

The method of topology is very clear.Then there's a question asking to use adjoint representation of lie group $SU(2)$ $(\operatorname{adj}:SU(2)\to GL(su(2)))$to prove this. I can't solve this .
3
votes
1answer
555 views

extending a vector field defined on a closed submanifold

Let M be a differentiable (smooth) manifold, and S a closed submanifold. Let X be a vector field on S. Prove that X is the restriction of a vector field Y defined on M. I tried this way but i'm not ...
3
votes
2answers
178 views

A smooth function f satisfies $\left|\operatorname{ grad}\ f \right|=1$ ,then the integral curves of $\operatorname{grad}\ f$ are geodesics

$M$ is riemannian manifold, if a smooth function $f$ satisfies $\left| \operatorname{grad}\ f \right|=1,$ then prove the integral curves of $\operatorname{grad}\ f$ are geodesics.
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1answer
148 views

Isometric embedding of the 2-sphere in $\mathbb{R}^3$

Can anyone give me a hint or a reference that would help to understand the following result : The only compact $\mathcal{C}^{\infty}$-submanifold of $\mathbb{R}^3$ of constant curvature ...
2
votes
1answer
361 views

Approximate expression for the metric in normal coordinates

In the Wikipedia article on Ricci curvature (here) it is mentioned that one can approximate the metric g in normal coordinates by \begin{equation} g_{ij} = \delta_{ij} - \frac{1}{3} R_{ikjl} \,x^kx^l ...
12
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1answer
313 views

Is it possible to formulate variational calculus geometrically?

In textbooks I've seen differential geometry is done with finite-dimensional manifolds. Is it possible to generalise to banach manifolds so as to formulate the calculus of variations within it, or ...
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1answer
239 views

First fundamental form

This is a practice exercise from a geometry textbook by P. Wilson. Suppose we have a Riemannian metric of the form $|dz|^2/h(r)^2$ on an open disc of radius $\delta>0$ centered on the origin in ...
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230 views

Gaussian Curvature

Can anyone see the connection between the Gaussian curvature of the ellipsoid $x^2+y^2+az^2-1=0$ where $a>0$ and the integral $\int_0^1  {1\over (1+(a-1)w^2)^{3\over 2}}dw$? I am guessing ...
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2answers
236 views

Charts, spheres and determinants

Here are some things I don't understand, I would be very grateful for any help! I am trying to find a chart on the unit sphere that preserves area. The most natural map that springs to my mind is ...
3
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1answer
265 views

How should I deal with this two-dimensional $\frac{0}{0}$ limit?

Here is my question: Does the following limit exist? $$ \lim_{y\to\xi}\frac{(\xi_i-y_i)(\xi_j-y_j)({{\xi}-y})\cdot n(y)}{|\xi-y|^5},\quad 1\leq i,j\leq 3,\tag{*} $$ where $S\subset{\mathbb ...
0
votes
1answer
158 views

Cartesian Product of the Real line with a discrete sets

Suppose $S$ is a set of n points, that is $|S| =n$ seen as a discrete smooth manifold. Then is the cartesian product of manifolds $\mathbb{R} \times S \simeq \mathbb{R}^n$? If not what is it?
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vote
1answer
48 views

Is it possible to achieve the following form?

Is it at all possible to, by a change of variables, transform the metric $dx^2+dy^2\over g(r)^2$ where $g$ is a function and $r=\sqrt{x^2+y^2}$ to something of the form $du^2+f(u,v) dv^2$? Thank you.
2
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1answer
166 views

Changing of variables from Cartesians to polars

How can we change variables from $(x,y)$ to $(r,\theta)$ for the metric on the open disc $r<\delta$ defined by $(dx^2+dy^2)\over g(\sqrt{x^2+y^2})^2$ where $g(\sqrt{x^2+y^2})>0$ $\forall ...
0
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1answer
71 views

classifying vortices whose base space is $S^{3}$ or $S^{7}$

On $S^{1}$ there are three choices for a vector pointed inward to the same angle at the tangent line at every point. The vector is either in the opposite direction as the normal, or to the left of ...
2
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1answer
174 views

Self-intersection of parametric surface using Gauss-Bonnet theorem

I am trying to detect when a closed parametric surface intersect itself. My surface is described as a triplet of parametric functions $x(u,v)$, $y(u,v)$ and $z(u,v)$ where $u,v\in[0,1]$. For that ...
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299 views

what does following matrix says geometrically

Let $M\subset \mathbb C^2$ be a hypersurface defined by $F(z,w)=0$. Then for some point $p\in M$, I've $$\text{ rank of }\left( \begin{array}{ccc} 0 &\frac{\partial F}{\partial z} ...
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43 views

Characterizing surfaces in $R^3$ in which every point is an umbilic point

How can it be shown that the only such surfaces are spheres or planes?
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1answer
42 views

How to show given PDE preserves length while evolving curve to circle?

I was given the PDE $C_t = (L/2\pi - 1/k)N(p,t) $ and $C(p,0) = C_0(p)$ where $k$ = curvature of the evolving curve, and $C(p,t)$ is the family of closed ...
3
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1answer
93 views

Calabi flow and Robinson - Trautman equation

Given a metric $g_{a\overline b}$ defined on a Kaheler manifold $K$, the Calabi flow is defined by the equation: $$\partial_u g_{a\overline b}=\frac{\partial^2 R}{\partial Z^a \partial Z^\overline ...
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1answer
1k views

Shape operator vs second fundamental form

Is the any difference between shape operator and second fundamental form for surfaces?
2
votes
2answers
372 views

How to go from local to global isometry

Let $M$ be a connected complete Riemannian manifold, $N$ a connected Riemannian manifold and $f:M \to N$ a differentiable mapping that is locally an isometry. Assume that any two points of $N$ can be ...
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1answer
122 views

Riemannian metric - basic question

I tried to google the following but couldn't find an answer that helped - so I hope I might find some here - the question is short and very basic (I guess) : what does it mean when someone writes ...
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1answer
197 views

The embedding of smooth manifold

I have run into a problem in my differential geometry book. Let $M$ be a smooth manifold and $F={C^\infty }(M,\mathbb R)$. Define a mapping $i:M \to {\mathbb R^F}$ by ${i_f}(x) = f(x)$ for $x \in M,f ...