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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61
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9answers
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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 ...
56
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4answers
6k views

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 ...
29
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2answers
3k 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 ...
27
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3answers
2k views

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 ...
5
votes
2answers
959 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 ...
26
votes
1answer
2k 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 ...
8
votes
2answers
296 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
262 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 ...
25
votes
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 ...
18
votes
2answers
871 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 $\theta_t$ ...
7
votes
2answers
590 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 ...
34
votes
3answers
4k 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 $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
8
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3answers
452 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 ...
6
votes
1answer
1k 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
votes
2answers
561 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
251 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
613 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 ...
17
votes
4answers
4k views

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 ...
43
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 ...
11
votes
3answers
998 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, ...
18
votes
9answers
4k views

Introductory texts on manifolds

I was studying some hyperbolic geometry previously and realised that I needed to understand things in a more general setting in terms of a "manifold" which I don't yet know of. I was wondering if ...
25
votes
4answers
719 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 ...
12
votes
5answers
905 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? ...
11
votes
2answers
2k views

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
774 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 $H^1\left(M,GL(n,\mathcal{C}^\infty_M)\right)$ ...
9
votes
2answers
884 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?
4
votes
1answer
583 views

What is the practical difference between abstract index notation and “ordinary” index notation

I understand that in "normal" index notation the indexes can be thought of as coordinates of scalar values inside a tabular data structure, while in the abstract index notation they can not. However, ...
4
votes
1answer
828 views

Problem book on differential forms wanted

I want to get used to differential forms. Thus I would like to solve a bunch of problems, especially on integration of differential forms. So I need a collection of problems with answers/solutions, ...
9
votes
2answers
627 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
3answers
106 views

A smooth function $f:S^1\times S^1\to \mathbb R$ must have more than two critical points.

I am trying to show that a smooth function $f:S^1\times S^1\to \mathbb R$ must have more than two critical points. Since $f$ attains maximum and minimum, it must have at least two critical points. ...
8
votes
1answer
386 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 ...
7
votes
2answers
454 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 ...
6
votes
2answers
821 views

Curvature of geodesic circles on surface with constant curvature

I am trying to solve the following exercise: Prove that on a surface of constant curvature the geodesic circles have constant curvature. "Constant curvature" in case of the surface I take to ...
5
votes
2answers
1k views

Proof that angle-preserving map is conformal

Let $\phi: S \to \bar{S}$ be a diffeomorphism between two surfaces in $\mathbb{R^3}$. Such a map is called conformal if for all $p \in S$, and $v_1, v_2 \in T_p(S)$ (the tangent plane) we have ...
4
votes
2answers
310 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
128 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 ...
5
votes
1answer
227 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, ...
2
votes
1answer
1k views

Prove that curve with zero torsion is planar

I have proved that a planar curve of zero curvature is a straight line. It follows from the Frenet equations. But now I need to prove that if $\varkappa=0$, then the space curve $\mathbf{r}(t)$ is ...
1
vote
1answer
67 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
219 views

gaussian and mean curvatures

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 ...
25
votes
10answers
4k views

“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
2answers
483 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. ...
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 ...
6
votes
1answer
819 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 ...
8
votes
4answers
18k 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 ...
14
votes
2answers
952 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 ...
7
votes
2answers
163 views

Smooth surfaces that isn't the zero-set of $f(x,y,z)$

The zero-set of any smooth function $f(x,y,z)$ with a non-vanishing gradient is a smooth surface. I was wondering if the reverse is true: is every smooth surface in $E^3$ the zero-set of some smooth ...
7
votes
2answers
935 views

Expression for Levi-Civita Connection

I'm having trouble with the following exercise in do Carmo's Riemannian geometry. Let $X$ and $Y$ be differentiable vector fields on a Riemannian manifold $M$. Let $p \in M$ and let $c: I \to M$ be ...