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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elements of the fundamental group and closed geodesics?

Let $(M;g)$ be a closed manifold . fix a point $x$ in $M$ and denote by $G=\pi_1(M;x)$ now let $\alpha$ in G i have 2 questions : 1- can $\alpha$ be represented by a closed geodesic ? 2- can ...
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235 views

Intersection points in a one-parameter family of lines

Given is a one-parameter family of lines, $$L(t) = \{ a(t) + \lambda w(t) : \quad \lambda \in \mathbb{R} \}$$ in which the base point $a$ and the direction vector $w$ vary smoothly with a parameter ...
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1answer
257 views

Euclidean isometries of $\mathbb{R}^3$ vs isometries of surfaces

Euclidean isometries in $\mathbb{R}^3$ are compositions of a translation and an orthogonal transformation. Each Euclidean isometry is a surface isometry that preserves length of rectifiable curves, ...
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1answer
401 views

Tangent vectors as derivations & differential map

I am a bit stuck about figuring out that the result of a differential map $f_{\ast p}$ defined on a tangent space is a derivation in the "other" tangent space. Suppose $f \colon N \to M$ is a map ...
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1answer
148 views

rotation vector

If $t(t),n(t),b(t)$ are rotating, right-handed frame of orthogonal unit vectors. Show that there exists a vector $w(t)$ (the rotation vector) such that $\dot{t} = w \times t$, $\dot{n} = w \times n$, ...
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499 views

Special orthogonal group as a manifold

Knowing that $\mathrm{O}(n,\mathbb{R})$ is a closed submanifold (of the general linear group) and that $\mathrm{SO}(n,\mathbb{R})$ is one of its subgroups with the same dimension, is there a quick way ...
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1answer
245 views

Orientability of the sphere

how does one explain the following: "The sphere can be covered by 2 open sets using stereographic projection in such a way that the intersection of these 2 sets is a connected set $W$.Let $p \in W$, ...
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1answer
280 views

Definition of tangent space

Terry Tao defines tangent space here as equivalence classes of continuously differentiable curves $\gamma : I \rightarrow G$ where $I$ is an open interval. On the other hand, Wikipedia defines it as ...
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342 views

Total Derivative and Multilinear Functions

So I'm starting to work through Spivak's Calculus on Manifolds and I'm having a little trouble verifying some of the claims made in the book problems. To review: Given a function ...
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1answer
99 views

On the level, is $w \perp \vec{v}$?

Take a curve $\vec{r} = \vec{r}(t)$ that stays on the level $w=c$ where $c$ is a constant. Velocity is $\vec{v} = \frac{d\vec{r}}{dt}$ and is tangent to the level $w=c$ because it's tangent to the ...
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1answer
135 views

Is any foliation on a 2-torus induced by a suitable flow?

Consider the 2-dimensional torus $T^2=\mathbb{R}^2/\mathbb{Z}^2$, and a foliation on it (for example a foliation in circles, maybe the partition of the torus obtained form a Hopf-related map). I'm ...
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396 views

Tangent Plane to a regular surface

I'd like some help with the following question: Find the equation of the tangent plane in $(x_0,y_0,z_0)$ to a regular surface given by $f^{-1}(0)$, where $0$ is a regular value. I tried to find ...
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1answer
133 views

Embedding a manifold in the disk

I don't understand a sentence made by Hirsch in his Differential Topology at page 175: If $k > n+1$ and $M^n = \partial W^{n+1}$, then an embedding $M^n \hookrightarrow S^{n+k}$ extends to a neat ...
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1answer
133 views

Do the maximum and minimum values of a Laplacian eigenfunction have the same magnitude?

Let $\Delta$ be the scalar Laplace-Beltrami operator on a compact, connected, orientable 2-manifold without boundary smoothly embedded in $\mathbb{R}^3$ and let $\phi$ be one of its eigenfunctions, ...
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101 views

About $\operatorname{Spin}^\mathbb{C}$ structures

Let $X$ be a orientable 4-manifolds. I know that $X$ can be endowed with $\operatorname{Spin}^\mathbb{C}$ structures by the choice of integral lift of $w_2(X)$ (second Stiefel-Whitney class of ...
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618 views

When aren't Christoffel symbols symmetrical with respect to its bottom indexes, and why?

When aren't Christoffel symbols symmetrical with respect to its bottom indexes? Why isn't the symmetry of second derivatives true in this case?
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166 views

Is the extra condition in this definition superfluous?

I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
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798 views

Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge Theory

I am looking for a textbook that might serve as an introduction to principal bundles, curvature forms and characteristic classes, and perhaps towards 4-manifolds and gauge theory. Currently, the only ...
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116 views

How to make a uniform grid of points in a curved space?

If a space has a differential volume element defined by: $d\Omega=\sin^2(\alpha)\sin(\theta)d\alpha d\theta d\phi$ And $\alpha \in [0,\pi/2]$, $\theta \in [0,\pi]$, and $\phi \in [0,2\pi]$ How can ...
2
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2answers
68 views

The tangency of two surfaces on a geodesic

If $S$ is a surface with a geodesic on it, can we find another surface $S'$ such that these surfaces are tangent on the geodesic with the additional condition that there is no other intersection? ...
3
votes
2answers
336 views

Calabi-Yau Manifolds

In short, I'm hoping for some reading recommendations. I'm starting to do some work with Calabi-Yau manifolds, though my prerequisites are fairly minimal in differential geometry. I've taken a ...
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184 views

How does one determine $n$-spheres of curvature?

I am aware of circles of curvature and I am simply wondering to what extent does this generalize to $n$-dimensions. Specifically, if some surface in $n$-dimensional space is represented ...
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120 views

Complex torus and integral of forms

Start with a complex torus $X$ of dimension $n$ and a basis $e_1,...,e_{2n}$ for its first integral cohomology. Let $f_1,...f_{2n}$ be the dual basis vectors, so they are in $H^{1}(X,\mathbb Z)^*$. ...
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1answer
309 views

Are integrations on forms “different” from Riemann integrations?

I was amazed by the power of integration on forms when I learned that the Stokes' theorem can be written in a beautiful way (don't assume that I know more than this fact itself): $$ ...
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615 views

Where can I read about elliptic operators on manifolds?

In Voisin's beautiful book on Hodge theory she gives a proof that every cohomology class can be represented by a harmonic form by referring to the the following theorem on elliptic operators: Let $P ...
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1answer
260 views

Partial derivative notation: is that a projection function?

Consider the following definition: Let $(U,\phi)$ be a chart and $f$ a $C^\infty$ function on a manifold $M$ of dimension $n$. As a function into $\mathbb{R}^n$, $\phi$ has $n$ components ...
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393 views

Singularities of a quotient of a smooth projective variety by a finite group

There is a nice smooth projective variety of high dimension, and a finite group G acting on it. Assume X/G exists. What can one say about the singularity of X/G? e.g. Are they always isolated? (I was ...
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1answer
225 views

transformation of vector fields under coordinate transformations

Consider two open subsets $\Omega, \Omega^{\prime}\subset \mathbb{R}^n$. Now consider a (volume preserving) diffeomorphism \begin{align} \varphi:\Omega^{\prime}\to\Omega; \alpha\to \varphi(\alpha) ...
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195 views

Associated bundle

Given a principal $G$-Bundle $P\rightarrow X$ and if we let $G$ act on itself by multiplication (denote this action by $\rho$) we obtain an associated bundle $P\times_{\rho} G=(P\times G)/\sim$ where ...
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1answer
67 views

Differentiable connections

Let $E$ a vector bundle on a differentiable manifold and $D: E\rightarrow E\otimes \Omega^1$ an homomorphism, with $\Omega^1$ differential 1-forms. If I take the map $D\wedge D$ which is the target ...
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1answer
365 views

The Dual Pairing

My understanding from the reading the Wikipedia article on Dual Pairs is that a dual pair is comprised of two vector spaces $X$ and $Y$ over a field $\mathbb{K}$ together with a nondegenerate ...
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1answer
305 views

Jacobian when representing integral of differential form by Riemann integral?

In Terence Tao's note: If $Ω$ is any open bounded domain in $R^n$ , we then have the identity $$\int_Ω f (x)dx_1 ∧ . . . ∧ dx_n = \int_Ω f (x) dx$$ where on the left we have an integral of ...
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1answer
302 views

rolling wheel problem

To achieve this: http://en.wikipedia.org/wiki/Square_wheel, what should $L$ be?
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1k views

understanding covariant derivative (connexion)

My lecturer defined the covariant derivative as in this section from Wikipedia: http://en.wikipedia.org/wiki/Covariant_derivative#Vector_fields. From this, he defines the operator $\nabla_X Y$ to mean ...
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1answer
93 views

Smoothness of the parallel surfaces

For the "class $C^n$", I use the following definition from Rainer Kress's Linear Integral Equations(2nd edition): A bounded open domain $D\subset{\mathbb R}^m$ with boundary $\partial D$ is said ...
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480 views

Diffeomorphisms: Examples and Importance

Can anyone give a specific example of a diffeomorphism and also of composing a function with a diffeomorphism and how this helps mathematics as a whole? In other words, how does this fit into the ...
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1answer
213 views

Conformal Flatness of 2-manifolds

So, due to the existence of isothermal coordinates, all 2-manifolds are conformally flat. The consequences of this are a bit confusing to me- this means one can conformally map, for instance, the ...
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votes
1answer
370 views

Meaning of restriction of a vector field on a submanifold

I'm trying to make sense of what restricting a vector field means. More specifically, if we have $S^3$ as a submanifold of $\mathbb{R}^4$ and the vector field, say, $X=(x_4,-x_2,x_3,-x_1)$, what does ...
4
votes
1answer
427 views

Dupin's indicatrix of the monkey saddle

The "monkey saddle" is a parametric surface defined by $$ \begin{eqnarray} x & = & u \\ y & = & v \\ z & = & u^3 - 3 v^2 u \end{eqnarray} $$ Its second fundamental form ...
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2answers
957 views

Are “differential forms” an algebraic approach to multivariable calculus?

I am recently learning some basic differential geometry. As I understand, differential forms provide a neat way to deal with the topics in calculus such as Stoke's theorem. In order to define the ...
2
votes
1answer
322 views

How to solve a system of dot products

I have the following system of simultaneous dot products in $\mathbb{R}^3$ which I am trying to solve for $x$: $$ \begin{eqnarray} x \cdot t & = & p \cdot t \\ x \cdot n & = & ...
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1answer
188 views

Compute the degree of map

Let $S:SU(2)\rightarrow SU(2)$ be defined as $S(X)=X^{4}$. Compute the degree of $S$. Now, $SU(2)$ is homeomorphic to $S^{3}$, so the degree can be taken as:$$ \int_{S^{3}}S^{*}\omega=(\deg ...
3
votes
1answer
143 views

Relation between a Lie group and Lie algebra representation for $W \otimes V$

We can define a representation of a Lie group and get the induced representation of the Lie algebra. Let $G$ act on $V$ and $W$, $\mathfrak{g}$ be the Lie algebra associated to $G$ and $X \in ...
5
votes
1answer
554 views

Degree of Gauss map equal to half the Euler characteristic and Poincaré-Hopf

The Poincaré-Hopf theorem states that for a smooth compact $m$-manifold $M$ without boundary and a vector field $X\in\operatorname{Vect}(M)$ of $M$ with only isolated zeroes we have the equality ...
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1answer
545 views

The integral of the mean curvature vector over a closed immersed surface

Suppose we have a closed, orientable, smooth surface $\Sigma$ immersed smoothly in $\mathbb R^n$ via $f:\Sigma \rightarrow \mathbb R^n$. Impose a Riemannian structure on $\Sigma$ by taking $g_{ij} = ...
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1answer
183 views

Surjection that increases dimensions

This question is somewhat inspired by a question on MathOverflow, but it is not necessary to read that question to understand what I am about to ask. It is well known that one can establish a ...
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203 views

On a remark in Foundations of mechanics, 2nd Edition, by Abraham and Marsden

Given a $2$-form $\omega$ on a manifold $M$, let us denote by $N$ the kernel of $\omega$, i.e. $N:=\{u\in TM : \omega(u,\cdot)=0\}$. Their Proposition 5.1.2 shows that if $\omega$ has constant rank ...
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1answer
1k views

A question on closed convex plane curves (from Do Carmo)

Let $\alpha (s)$ , $s\in [0,l]$, be a closed convex plane curve positively oriented. The curve $\beta(s)=\alpha (s) -rn(s)$, where $r$ is a positive constant and $n(s)$ is the normal vector, is called ...
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1answer
575 views

local property of a curve to deduce a global property

I have a question about some facts about curves. Consider a curve such that all the normals intersect at a fixed point. Prove that the trace of the curve it's contained in a circle. And prove that if ...
4
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1answer
424 views

Wald's definition of parallel transport

I was unsure whether to ask this here or at a physics SE. Wald's "General Relativity" defines parallel transport as follows: $\nabla$ is a derivative operator (is linear, obeys Leibniz rule, ...