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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Teaching myself differential topology and differential geometry

I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for ...
51
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4answers
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Why is a circle in a plane surrounded by 6 other circles

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other number? I'm ...
25
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3answers
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Why are smooth manifolds defined to be paracompact?

The way I understand things, roughly speaking, the importance of smooth manifolds is that they form the category of topological spaces on which we can do calculus. The definition of smooth manifolds ...
3
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2answers
740 views

About connected Lie Groups

How can I prove that a connected Lie Group is generated by any neighborhood of the identity? The result is almost trivial for $R^n$ but I tried using the open subgroup generated by this ...
7
votes
2answers
248 views

Question about Angle-Preserving Operators

This an exercise out of Spivak's "Calculus on Manifolds". Edit: There was a typo in the exercise as is noted below in the answers. The statement has been edited to reflect this. Given ...
7
votes
1answer
258 views

Finding a smooth function less than some given (positive) continuous function

Let $M$ be a smooth manifold ($dim\ge 1$). Let $f:M\to\mathbb{R}$ be a positive continuous function. Prove there is a smooth map $g\in C^{\infty}(M)$ such that $0<g<f$. I knew this would ...
23
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3answers
1k views

Roadmap to study Atiyah-Singer index theorem

I am a physics undergrad and want to pursue a PhD in Math (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non ...
28
votes
2answers
2k views

Which manifolds are parallelizable?

Recall that a manifold $M$ of dimension $n$ is parallelizable if there are $n$ vector fields that form a basis of the tangent space $T_x M$ at every point $x \in M$. This is equivalent to the tangent ...
18
votes
2answers
770 views

Relationship between the zeros of a vector field and the fixed points of its flow

I'm having a little trouble here and would appreciate some hints. Let $M$ be a compact manifold without boundary and let $X$ be a smooth vector field on $M$ with only isolated zeros. Let $\theta_t$ ...
32
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3answers
3k views

Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
7
votes
2answers
496 views

Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms? I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
6
votes
1answer
917 views

Conformal transformation of the curvature and related quantities

Suppose we have a Riemannian manifold ${(M,g)}$, where ${g}$ is the metric of ${M}$. If ${f}$ ${\in}$ ${D(M)}$ (i.e. smooth function on ${M}$), and ${f}$ is positive. So, we can define a new metric ...
14
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2answers
523 views

Is $M=\{(x,|x|): x \in (-1, 1)\}$ not a differentiable manifold?

Let $M=\{(x,|x|): x \in (-1, 1)\}$. Then there is an atlas with only one coordinate chart $(M, (x, |x|) \mapsto x)$ for $M$. We don't need any coordinate transformation maps to worry about ...
4
votes
1answer
1k views

Properly Defining a Smooth Curve

I have seen many different definitions of what it means for a curve to be "smooth". In this question, for instance, a curve $\gamma \colon [a,b] \longrightarrow \mathbb{R^n}$ is defined to be smooth ...
4
votes
1answer
202 views

surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism

Does there exist a surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism? I tried to modify $\exp: \mathbb{C} \to \mathbb{C}$ to be surjective, but I find it hard to ...
3
votes
2answers
563 views

An application of partitions of unity: integrating over open sets.

In Spivak's "Calculus on Manifolds", Spivak first defines integration over rectangles, then bounded Jordan-measurable sets (for functions whose discontinuities form a Lebesgue null set). He then uses ...
41
votes
1answer
2k views

What is the solution to Nash's problem presented in “A Beautiful Mind”?

I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve ...
24
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1answer
1k views

Is there an easy way to show which spheres can be Lie groups?

I heard that using some relatively basic differential geometry, you can show that the only spheres which are Lie groups are $S^0$, $S^1$, and $S^3$. My friend who told me this thought that it ...
24
votes
4answers
694 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 ...
7
votes
3answers
417 views

$f^*dx_i = \sum_{j=1}^l \frac{\partial f_i}{\partial y_j} dy_j = df_i$

Guillemin and Pollack's Differential Topology Page 164: $U \subset \mathbb{R}^k$ and $V \subset \mathbb{R}^l$ be open subsets. Let $f: V \to U$ to smooth. Use $x_1, \dots, x_k$ for the standard ...
12
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5answers
852 views

Why the interest in locally Euclidean spaces?

A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds). What is the special feature of Euclidean spaces that makes them interesting? ...
10
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2answers
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Definitions of Hessian in Riemannian Geometry

I am wondering is there any quick way to see the following two definitions of Hessian are coinside with each othere without using local coordinates? $\operatorname{Hess}(f)(X,Y)= \langle \nabla_X ...
7
votes
3answers
659 views

Vector bundle transitions and Čech cohomology

I have read that transition maps $g_{\alpha\beta}:U_\alpha\cap U_\beta\to GL(n)$ of a vector bundle of rank $n$ are related to the Čech cohomology group $H^1\left(M,GL(n,\mathcal{C}^\infty_M)\right)$ ...
7
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2answers
729 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?
9
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2answers
607 views

Are there higher-dimensional analogues of sectional curvature?

I recently learned that on Riemannian manifolds, one can define the sectional curvature (http://en.wikipedia.org/wiki/Sectional_curvature) of a (2-dimensional) plane section. I was wondering if a ...
8
votes
1answer
340 views

embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$

Consider the classic map $$F:\mathbb{RP}^2\rightarrow \mathbb{R}^4$$ defined by $$F[x,y,z]=(x^2-y^2,xy,xz,yz)$$. This defines a smooth embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$. It is clearly ...
3
votes
3answers
232 views

Topological boundary vs geometric boundary

Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$ $M_2=\{(x,y) \mid x^2+y^2\le1\}$ What are the interior of $M_1$ and $M_2$ ? And What are the boundary of $M_1$ and $M_2$ ? How to find them? ...
4
votes
2answers
251 views

Are there any compact embedded 2-dimensional surfaces in $\mathbb R^3$ that are also flat?

Let $\overline{g}$ be the flat metric on $\mathbb{R}^3$. I would like to know if there is any compact embedded 2-dimensional surface $M$ in $\mathbb{R}^3$ (without boundary) such that ...
2
votes
1answer
96 views

Differentiating the determinant of the Jacobian of a diffeomorphism (don't understand a proof)

For each $t$, let $A_t:\Omega_0 \to \Omega_t$ be a bi-Lipschitz map between open sets in $\mathbb{R}^n$. The map is also invertible. It satisfies $$\frac{d}{dt}A_t(y) = w(A_t(y),t)$$ where $w$ is a ...
0
votes
2answers
88 views

Lie bracket in local coordinates. Find the formula $c^{k}$ in terms of $a^{i}$ and $b^{j}$

This is from T.U Loring's manifold book. I tried. But I didnt do the question. Please show me how to solve instructively and explicitly. I want to learn this topic. Thank you for help.
0
votes
1answer
73 views

The differential $i∗ : TpS_{2} → TpR_{3 }$ maps $ ∂/∂u|p,∂/∂v|p $ into $TpR_{3}. $ Find $(α_{i}, β_{i}, γ_{i})$

Hi! This was my homework. Prof. sent its answer. But I didnt understand how can this answer be reached? Please can someone explain this?
5
votes
1answer
208 views

On minimizing the area of an enclosing surface subject to nonnegative Gaussian curvature

This is inspired by this previous question on physical processes that might give rise to convex hulls. Consider the problem of gift-wrapping a three-dimensional object using an inextensible material, ...
1
vote
1answer
64 views

Decomposition of cohomology group on $S^{n}$

If we have decomposition of cohomology group on $S^{n}$ it looks like $H^{n}(S^{n})=H^{n}(S^{n})_{+}\oplus H^{n}(S^{n})_{-}$, where $H^{n}(S^{n})_{\pm}$ cohomology of invariant or anti-invariant $n$ ...
1
vote
1answer
167 views

gaussian mean curvature

I am trying to review, and learn about how to compute and gaussian and mean curvature. Given $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$, how can I compute the gaussian and mean ...
23
votes
10answers
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“Immediate” Applications of Differential Geometry

My professor asked us to find and make a list of things/facts from real life which have a differential geometry interpretation or justification. One example is this older question of mine. Another ...
37
votes
4answers
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How to identify surfaces of revolution

Given a surface $f(x,y,z)=0$, how would you determine whether or not it's a surface of revolution, and find the axis of rotation? The special case where $f$ is a polynomial is also of interest. A ...
22
votes
1answer
389 views

Why is the Laplacian important in Riemannian geometry?

As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why. ...
11
votes
3answers
745 views

Geometrical interpretation of Ricci curvature

I see the scalar curvature $R$ as an indicator of how a manifold curves locally (the easiest example is for a $2$-dimensional manifold $M$, where the $R=0$ in a point means that it is flat there, ...
13
votes
2answers
1k views

Geometric interpretation of connection forms, torsion forms, curvature forms, etc

I have just begun learning about the connection 1-forms, torsion 2-forms, and curvature 2-forms in the context of Riemannian manifolds. However, I am finding it hard to relate these notions to any ...
15
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3answers
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What is a covector and what is it used for?

From what I understand, a covector is an object that takes a vector and returns a number. So given a vector $v \in V$ and a covector $\phi \in V^*$, you can act on $v$ with $\phi$ to get a real number ...
8
votes
3answers
13k views

Surface Element in Spherical Coordinates

In spherical polars, $$x=r\cos(\phi)\sin(\theta)$$ $$y=r\sin(\phi)\sin(\theta)$$ $$z=r\cos(\theta)$$ I want to work out an integral over the surface of a sphere - ie $r$ constant. I'm able to derive ...
6
votes
1answer
580 views

Are there simple examples of Riemannian manifolds with zero curvature and nonzero torsion

I am trying to grasp the Riemann curvature tensor, the torsion tensor and their relationship. In particular, I'm interested in necessary and sufficient conditions for local isometry with Euclidean ...
14
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2answers
837 views

The space of Riemannian metrics on a given manifold.

For a finite-dimensional smooth (Hausdorff, second-countable) manifold $M$, consider the set $$\mathcal{Met}(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$ I'd like to know about the typical ...
5
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4answers
183 views

Reason for Continuous Spectrum of Laplacian

For the circle $S^1$, it is well-known that the Laplace-Beltrami operator $\Delta=\text{ div grad}$ has a discrete spectrum consisting of the eigenvalues $n^2,n\in \mathbb{Z}$, as can be seen from the ...
5
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0answers
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 ...
9
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2answers
909 views

On surjectivity of exponential map for Lie groups

A recent question made me realize I didn't know any proof that exponential of a Lie algebra $\mathfrak g$ of a compact connected Lie group $G$ is surjective. After a bit of thinking I've come up with ...
6
votes
3answers
3k views

Shortest proof for 'hairy ball' theorem

I want to make a project at differential geometry about the Hairy Ball theorem and its applications. I was thinking of including a proof of the theorem in the project. Using the Poincare-Hopf Theorem ...
6
votes
2answers
2k views

Classsifying 1- and 2- dimensional Algebras, up to Isomorphism

I am trying to find all 1- or 2- dimensional Lie Algebras "a" up to isomorphism. This is what I have so far: If a is 1-dimensional, then every vector (and therefore every tangent vector field) is ...
4
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3answers
684 views

meaning of dual space of a tangent space?

We know that a tangent vector is a directional derivative operartor, and the collection of all tangent vectors at a point is a tangent space. I don't understand the intuitive meaning behind the dual ...
3
votes
2answers
433 views

A curve parametrized by arc length

Let $C$ be a plane curve parametrized by arc length by $\alpha(s)$, $T(s)$ (unit tangent vector) and $N(s)$ (unit normal vector). Prove that $$\frac{d}{ds} N(s)=-\kappa(s)T(s).$$ I know that ...