Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

32835 questions
12
votes
3 answers

Showing a vector field is tangent to the 2-sphere

How would I go about showing that the vector field $X = x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x}$ is tangent to the unit sphere in $\mathbb{R}^3$? I can see it pictorially but I'm not sure what would be required to actually prove…
Anna
  • 463
12
votes
2 answers

When is an $n$-dimensional manifold characterized by its $m$-dimensional submanifolds?

For which $m$, $n$ (if any) is the following true: if $M$ and $M'$ are smooth manifolds of dimension $n$, and $\Phi$ is a bijection from $M$ to $M'$ such that for any subset $S$ of $M$, $\Phi(S)$ is an embedded submanifold of $M'$ of dimension $m$…
Cian
  • 696
12
votes
1 answer

An irregular ball rolling on a plane, if know the path on ball surface, how to find the path on the plane?

An irregular ball has its local radius or curvature different at any surface point. It has pure rolling movement on the plane P. Several questions here: If I know the path curve from A to B on the ball, how to know the path on the plane, and vice…
george andrew
  • 245
12
votes
1 answer

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 $f:U\subset\mathbb{R}^n\rightarrow\mathbb{R}^n$ is one-to-one and…
Lost In Math
  • 717
12
votes
2 answers

Definition of the Lie coalgebra

I don't understand how the Lie coalgebra is defined. The literature is never really explicit in how it is constructed. So I was wondering if anybody could supply me with a simple example of how the Lie coalgebra is constructed. Let's say for…
Novo
  • 1,423
12
votes
2 answers

Why first fundamental form?

Here is an excerpt from the notes we are using: The first fundamental form dictates how one computes dot products of vectors tangent to the surface assuming they are expanded according to the basis $\frac{\partial q}{\partial u},\frac{\partial…
3x89g2
  • 7,542
12
votes
4 answers

Every manifold is locally compact?

Theorem. Every Manifold is locally compact. This is a problem in Spivak's Differential Geometry. However, don't know how to prove it. It gives no hints and I don't know if there is so stupidly easy way or it's really complex. I good example is…
simplicity
  • 3,694
12
votes
1 answer

Turning number VS winding number

To avoid confusion, here are the definitions of the objects in this question: 1) Let $\gamma:S^1\to\mathbb{R}^2\setminus\{0\}$ a smooth loop. The winding number of $\gamma$ is the number of times $\gamma$ encircles $0$. Similarly, if…
Amitai Yuval
  • 19,308
12
votes
2 answers

Local diffeomorphism from $\mathbb R^2$ onto $S^2$

Is there any local diffeomorphism from $\mathbb R^2$ onto $S^2$?
fiorerb
  • 642
11
votes
1 answer

Understand Cotangent Space as an Equivalence Class

In my differential geometry class, my teacher defined the co-tangent space as follows. Let $M$ be a smooth manifold and $p$ is a point on $M$. Now define two sets of $C^\infty$ real-valued functions defined on $M$. $\mathcal I_p := \left\{f\in…
LaTeXFan
  • 3,548
11
votes
3 answers

What is the difference between a Germ and a 1-form

The germ captures the local behaviour of a function at a point of a topological space. I'd like to know how it is different from a 1-form or its relationship with a 1-form. Pardon me if its a naive question in case its too simple please give a link…
Rajesh D
  • 4,247
11
votes
2 answers

A vector field is a section of $T\mathcal{M}$.

By definition, a vector field is a section of $T\mathcal{M}$. I am familiar with the concept of vector field, as well as tangent plane of a manifold. But such definition is not intuitive to me at all. Could some one give me some intuition? Thank you…
WishingFish
  • 2,412
11
votes
2 answers

Is the surface of a sphere and a crayon the same manifold?

In Schutz, Geometrical Methods of Mathematical Physics book, pg. 29, it was said that the sphere $S^2$ and the surface of a crayon has the same global structure: I also read that two spaces are the 'same' as manifolds if they are diffeomorphic. But…
TaeNyFan
  • 751
11
votes
1 answer

Closed space curves of constant curvature

Which closed space curves of constant curvature are there? Two families spring to mind: circles and closed helices around a torus (both having - in addition - constant torsion). What other families of curves of constant curvature are there? Are…
Hans-Peter Stricker
  • 18,159
11
votes
1 answer

Exercise in Do Carmo's "Riemannian Geometry": the Möbius band is nonorientable.

Of course there are many ways to prove this. However, I came across the following exercise (Ch. 0 #3). Prove that: (a) a regular surface $S\subset \mathbb{R}^3$ is an orientable manifold if and only if there exists a differentiable…
Aaron Mazel-Gee
  • 8,039
1 2 3
…
99 100