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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Local triviality of principal bundles

Suppose I define a principal $G$-bundle as a map $\pi: P \to M$ with a smooth right action of $G$ on $P$ that acts freely and transitively on the fibers of $\pi$. Does it follow that $P$ is locally ...
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421 views

Smooth curve with no Frenet frame

Let $\gamma: I \rightarrow \mathbb{R}^n$ be a smooth curve. We define a Frenet frame to be an orthonormal frame $X_1, \ldots X_n$ such that for all $1 \leq k \leq n$, $\gamma^{(k)}(t)$ is contained in ...
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1answer
128 views

The unit ball fibration in a tangent bundle

Let $X$ be a complex manifold equipped with a smooth hermitian metric $h$. We can define a sub-fibration $B \to X$ of the tangent bundle $T_X$ by requiring that the fiber over a point be the unit ball ...
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460 views

What is $T\mathbb{S}^2$?

I recently learned that the only parallelizable spheres are $\mathbb{S}^1$, $\mathbb{S}^3$, and $\mathbb{S}^7$. This led me to wonder: What is $T\mathbb{S}^2$? Is it diffeomorphic to a more ...
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1answer
107 views

System of inequalities

All functions are smooth and continuous. I use ' to signfy $d/dt$. Given: $q_i'(0)=0$, for i=1,2,3,4. $q_2(t)q_3(t)>q_1(t)q_4(t)$ for $0<=t<T$. T is finite. At t=T the inequality breaks. ...
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1answer
360 views

Umbilical points and maxima/minima of a function

Premise: I'm a bit unsure about the terminology most frequently used, so I'll try to be as clear as possible, please tell me if something isn't clear. Talking about regular smooth manifolds, by which ...
9
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1answer
707 views

Spivak and Invariance of Domain

On p.3 of the first volume of Spivak's Comprehensive Introduction to Differential Geometry, he says that it is an "easy exercise" to show that the invariance of domain theorem (if ...
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2answers
461 views

The Dido problem with an arclength constraint

It is well known that the solution to the classical Dido problem is a semicircle, and that the solution to the classical isoperimetric problem is a circle. It's also reasonably obvious that the ...
7
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1answer
258 views

Elementary question in differential geometry

I am trying to learn differential geometry (i.e., teach myself!) So here is a question that came up. For some $h > 0$, consider the cone $C_h = \{ (x,y,z) \; : \; 0 \le z = \sqrt{x^2 + y^2} ...
5
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1answer
219 views

Extension of Riemannian Metric to Higher Forms

I've been reading about Riemannian manifolds, and have come across a comment that says that for a metric $g$ on an $N$-dimensional manifold $M$, considered as a bilinear map $$ g:\Omega^1(M) \times ...
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1answer
1k views

Derivation of Basic Level Set Equations

For the level set method, $\phi(\vec{x},t)$ is the level set function in 3D and the level set $\phi(\vec{x},t) = 0$ forms the interface. For evolving $\phi$ the derivation says to imagine a particle ...
5
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405 views

is $\nabla \cdot ( c^2 \nabla)$ a Laplace-Beltrami operator?

I asked this on mathoverflow, and was suggested to ask it here. Someone mentioned, in passing, to me that $u \mapsto \nabla \cdot ( c^2 \nabla u)$ is a Laplace-Beltrami operator. Does anyone have ...
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0answers
320 views

Boundedness of the Christoffel symbols of a connection on the normal bundle [closed]

I have the following setting: Let (M,g) be a Riemannian manifold and $\iota: M \to R^N$ some isometric embedding. This means especially that the connection $\nabla^{TM}$ of M is given by the ordinary ...
0
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1answer
410 views

Relation between metric tensor and second fundamental form

I'm confused with these definitions. The metric of certain space and the second fundamental form seem to be the same object. I don't know what else to say, this is a pretty straight forward question. ...
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153 views

costructing a diffeomorphism

Let $i:N\to M$ be a smooth embedding, $\pi:E\to N$ a vector bundle and $s_0:N\to E$ is its zero section. I have an open neighborhood $U$ of $s_0(N)$ in $E$, and $f:U\to M$ is a smooth map such that ...
4
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0answers
206 views

Strange use of differentials - is $d{\bf x} \cdot d{\bf x}$ a dot product?

If ${\bf x}(s)$ is a curve in $\mathbb{R}^3$ on a surface parameterized by its arc length $s$, and ${\bf N}$ is the surface normal at ${\bf x}$, consider the following equality (with "$\cdot$" being ...
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2answers
353 views

Tangent Bundle and its (Isomorphic?) Dual Bundle

In general it is not true that a vector bundle $E$ is isomorphic to its dual bundle $E^*$. But it is true when the vector bundle is the tangnet space of a manifold (at least I think it's true). How ...
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273 views

Is the dual covector field of a nonzero vector field closed?

Given a nowhere vanishing vector field, say $E_1$, on a manifold $M^n$, it should be possible to extend this locally to a basis of vector fields $E_1,\ldots , E_n$ so that $E_i = ...
3
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3answers
178 views

Motivating differentiable manifolds

I'm reading lectures for the first time starting next week ^_^ The subject is calculus on manifolds I was told that the students I'll be lecturing are not motivated (at all), so I need to kick off the ...
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2answers
141 views

Relation between homogeneous spaces and principal bundles

What is the relation between homogeneous spaces and principal bundles. I've been reading the two definitions and am left confused as to whether one is a subset of the other or whether no such relation ...
3
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1answer
285 views

Compact manifold/Morse theory

I have a question concerning the proof of theorem 3.5 in Milnor's Morse Theory. This theorem states that if $f$ is a differentiable function on a Manifold M with no critical points, and if each $M^a ...
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1answer
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A proof of the Isoperimetric Inequality - how does it work?

Here is a nice proof of the isoperimetric inequality (equality part ommited): Isoperimetric Inequality If $\gamma$ is any simple closed piecewise $C^1$ curve of length $l$, with it's interior having ...
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4answers
824 views

Tangent Bundle on S^3

how to show T(S^3) isomorphic to S^3 cross R^3? so can I say it for every odd dimension?I have shown it for n=1
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5answers
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Is this an Accurate General description of Line Bundles?

As a newbie to vector bundles, it seems like all vector bundles I have run into ( not that many, I admit) need only two charts to be trivialized; one of these charts will contain the "trouble ...
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1answer
328 views

extension/“globalization” of inverse function theorem

I am curious as to what changes do we need to make to the hypotheses of the inverse function theorem in order to be able to find the global differentiable inverse to a differentiable function. We ...
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901 views

How to Visualize points on a high dimensional (>3) Manifold?

Are there any ways to visualize(plot/draw) points on a high dimensional (ex: dimension = 5) manifold?
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2answers
551 views

$C^\infty$ vs. $C^\omega$ surfaces

I would appreciate it if someone could explain the difference(s) between a $C^\infty$ and a $C^\omega$ surface embedded in $\mathbb{R}^3$. I ran across these terms in M. Berger's Geometry Revealed ...
2
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1answer
398 views

Maurer-Cartan 1- form as a connection 1-form

I'm trying to decipher a differential geometric comment on page 23-24 of Berline, Getzler, and Vergne's "Heat Kernels and Dirac Operators". Take a trivial vector bundle $E \times M$ in a manifold $M$ ...
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3answers
949 views

Vector Bundles Over a Manifold

Show that if a manifold $M$ is contractible, then every vector bundle over $M$ is equivalent to the trivial bundle. Got this as homework but I'm kind of lost in the hole vector bundle subject, so if ...
1
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1answer
170 views

Completely Geodesic

I'm having trouble showing the following implication: Let $M$ be a Riemannian manifold, let $L\subset M$ be a submanifold such that the following holds: If $\gamma: I \to M$ is a geodesic s.t. ...
3
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1answer
454 views

Nonpositive curvature, Theorem of Cartan-Hadamard

In my differential geometry course we had the following Theorem (Cartan-Hadamard): Let $M$ be a connected, simply connected, complete Riemannian manifold. Then the following are equivalent: $M$ has ...
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846 views

Ways in which a manifold can be geodesically incomplete

Naively I would have thought that a manifold becomes geodesically incomplete if there are missing points in it or if the geodesics are hitting a boundary. But I am not sure how to think of geodesic ...
25
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4answers
722 views

Is every Compact $n$-Manifold a Compactification of $\mathbb{R}^n$?

I read the result that every compact $n$-manifold is a compactification of $\mathbb{R}^n$. Now, for surfaces, this seems clear: we take an n-gon, whose interior (i.e., everything in the n-gon except ...
4
votes
5answers
818 views

Differential Geometry of curves and surfaces: bibliography?

Dear all, next year, I will probably teach a one-semester course of Differential Geomtry of curves and surfaces. Its content must be something along the lines of the first four chapters of Do Carmo's ...
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1answer
430 views

Symmetricity of the extrinsic curvature tensor

I had referred to this structure earlier in a previous question which went unanswered. If $u$ and $v$ are in $T_pM$ and $n \in (T_pM)^\perp$ then the extrinsic curvature form $K$ be defined as, ...
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1answer
630 views

How to find equation of cone's generatrix?

Given canonical cone equation, how to find equation of cone's generatrix?
0
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1answer
117 views

Optimization of gaussian curvature

What is the geometrical "meaning" of gaussian curvature?
1
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3answers
361 views

Parametric equations of curves

Is there a way to produce parametric equations for a curve?(If we do know cartesian coordinates of course)
4
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1answer
299 views

How does one characterize surfaces with constant nonzero Gaussian and mean curvature

I know that for any surface, the Gaussian curvature $K$ and mean curvature $H$ satisfy the inequality $H^2 \geq K$ , and the sphere is a surface where that inequality becomes an equation. Thus, the ...
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1answer
660 views

Stochastic interpretation of Einstein Equations

Einsteins theory of gravitation, general relativity, is a purely geometric theory. In a recent question I wanted to know what the relation of Brownian Motion to the Helmholtz equation is and got a ...
12
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1answer
398 views

Diffeomorphic, group-isomorphic Lie groups that are not isomorphic as Lie groups

Do there exist two Lie groups which are diffeomorphic as smooth manifolds, have isomorphic group structures, yet are not isomorphic as Lie groups? Of course, for this to happen, any diffeomorphism ...
6
votes
1answer
405 views

Relationship between Riemannian Exponential Map and Lie Exponential Map

It is well known that for a matrix Lie group the Lie exponential map is $e ^z$. This maps a tangent vector $z$ at the identity to a group element. On the other hand the general Riemannian ...
11
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1answer
505 views

Coordinate free proof that curvature is the “square” of the connection

Here's the setup. Consider a vector bundle $E$ over a manifold $M$ and let $\Omega^*(M, E)$ denote the space of $E$-valued differential forms (i.e. the space of sections of the vector bundle ...
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2answers
826 views

Integral curves of the gradient

Let $f : M \rightarrow \mathbb{R}$ be a differentiable function defined on a riemannian manifold. Assume that $| \mathrm{grad}f | = 1$ over all $M$. Show that the integral curves of $\mathrm{grad}f$ ...
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1answer
2k views

Intuitive explanation of Left invariant Vector Field

Intuitively what is meant by a left invariant vector field on a manifold?
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275 views

Riemannian Immersions into Euclidean Space?

The Whitney embedding theorem states that any smooth manifold can be embedded in Euclidean space. In the Riemannian setting this naturally leads to the question whether this can be done in such a way ...
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Curvature 0 and involutive horizontal distributions

I am trying to check why curvature 0 implies that the horizontal distribution is involutive. Let $\pi:P\to U$ be a principal $G:=GL_n$ bundle. Assume that $P$ is trivial and $\pi$ admits a section. ...
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1answer
171 views

Embed an $n\times n$ matrix into $R^{n^2}$

How to embed a matrix, for example, a $3x3$ singular matrix in to $R^9$? How to compute the induced metric? Is it just the Frobenius norm of the matrix? Many Thanks. sam
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2answers
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Help with definition of n-dimensional smooth manifold

Again, I am reading this. I am finding it a bit difficult to understand the definition of n-dimensional smooth manifold. Now, $\{U_a; x^1_a, x^2_a, ..., x^n_a\}$ ...
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1answer
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How do I compute Gaussian curvature in cylindrical coordinates?

I just asked this question on ask.metafilter, and it was suggested that I ask here. Though I'm talking about coding something up, this question is about the math behind it, not the implementation. ...