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
5
votes
1 answer

Given a smooth map Φ:M→M , does there always exist a proper submanifold S such that Φ(S)=S?

As opposed to tracking any individual point, what I'm concerned with is a manifold that occupies the same set of points as its image under the smooth map. Suppose $\Phi(x,y)=(x+1,y)$. Consider an arbitrary horizontal line $y=a$. While no individual…
Simon M
  • 657
5
votes
1 answer

Mercator Projection.

I am trying to derive the metrics for the Mercator Projection using standard spherical coordinates $f(\phi, \theta) = (\cos\phi \cos\theta, \sin\phi \cos\theta, \sin\theta) $. For simplicity, I consider the radius of the Earth is equal to 1. Then…
Member1434
  • 137
  • 6
5
votes
3 answers

What is the "typical" map between surfaces?

I've seen the concept of map between surfaces $S\to \overline{S}$ and at first, I was extremely confused because I spent a lot of time thinking on how to actually send points from $S$ to $\overline{S}$ "directly". I'll try to explain what I mean…
Red Banana
  • 23,956
  • 20
  • 91
  • 192
5
votes
1 answer

How restricted is the curvature of a manifold by choice of topology keeping the underlying set fixed?

Suppose we have a topological manifold, then by varying the connection on the manifold, we can varying the properties related to curvature of the manifold as curvature is solely a property of the connection. This led me to wonder, is there a precise…
tryst with freedom
  • 11,538
5
votes
1 answer

Tensorfields and Multilinear maps correspondence

So we defined a (r,s) tensor field $w$ on a smooth manifold $M$ as a choice of (r,s) tensors for every point $p \in M$ living over the respected tangent spaces. So for every $p \in T_pM$, $w(p)$ is an Element of $\bigotimes^r T_pM \otimes…
QED
  • 104
5
votes
1 answer

Question about definition of normal bundle from the quotient space

According to Wikipedia, the definition of normal bundle is defined as, Defintion $1$. [Normal bundle] Let $(M,g)$ be a Riemannian manifold, and $S\subset M$ a Riemannian submanifold. For a given $p \in S$ , a vector $n \in T_pM$ to be normal to…
Hanwoong Cho
  • 885
5
votes
2 answers

How to understand "maximal" in the definition of differentiable structure

Consider the definition of differentiable structure (Lectures on Differential Geometry, S.S. Chern): Suppose $M$ is an m-dimensional manifold. If a given set of coordinate charts ${\mathcal A} = \{(U,\phi_U),(V,\phi_V),(W,\phi_W),\cdots\}$ on $M$…
user9464
5
votes
2 answers

Shortest path in conformal maps of a surface

My intuition tells me that the shortest distance between two points on the surface corresponds to a line segment joining the two points on the map of said surface, because, the path on the surface is same as the shortest path in the map. However,…
tryst with freedom
  • 11,538
5
votes
0 answers

Decomposition of generalized Einstein condition into irreducible representations of orthogonal group

I struggle with making sense of the claim in Chapter 16.2 of Arthur Besse's book "Einstein manifolds". Let $\nabla$ be an affine connection on the manifold $\mathcal{M}$. We want to decompose the covariant derivative of the Ricci tensor $\nabla Ric$…
Ovserger
  • 61
5
votes
0 answers

Given a covariant derivative, is there a metric tensor that has the covariant derivative as the metric connection?

In differential geometry or general relativity, we usually think of the metric tensor $g_{\mu \nu}$ first and then introduce the metric connection. However, I wonder if we can go reverse. That is, let $M$ be a smooth manifold equipped with some…
Keith
  • 7,673
5
votes
1 answer

If a manifold admits a real analytic structure then the manifold is analytic?

I have been reading the notion of real analytic space in nLab and found a statement that puzzles me. That Whitney embedding theorem shows that every paracompact smooth manifold admits a real analytic structure. Whitney embedding theorem shows that a…
jaogye
  • 963
5
votes
2 answers

Global conformally flat coordinates in 2d spacetimes

Let $(M,g)$ be a 2 dimensional pseudo-Riemannian manifold that is topologically a disc. Is it possible to construct a global coordinate system in which the metric is conformally flat? I.e. coordinates $(t,x)$ which cover the whole manifold such that…
Dionigi
  • 83
5
votes
4 answers

What is a manifold on a Euclidean space?

In this semester I study differential geometry and in this chapter we want to define what is a surface. In order to do that we first define what a manifold is on a Euclidean space, not generally what is a manifold, and Euclidean space I mean…
領域展開
  • 2,139
5
votes
2 answers

Differential Geometry - Computation Help

I'm trying to learn differential geometry through one of MIT's online courses (lecture notes found here: http://ocw.mit.edu/courses/mathematics/18-950-differential-geometry-fall-2008/lecture-notes/ch1_revised.pdf) and am stuck with what should be…
user71216
  • 81
5
votes
2 answers

Surface with non-zero mean curvature means orientable

Let $M$ be a surface in $\Bbb R^3$ with non-zero mean curvature for every point. How could I show that this implies that $M$ is orientable? By our definition, orientable means that an unitary, normal vector can be defined continuously for every…
DashDotDashDot
  • 233
1 2 3
…
99 100