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
6
votes
1 answer

Is it known whether $S^6$ is a Kähler manifold?

I have just started to learn about Kähler manifolds and I now am wondering: Is it known whether $S^6$ is a Kähler manifold? By definition a Kähler manifold has 3 structures: a symplectic, a complex and a Riemannian structure. I know that for $n>3$…
user174981
  • 407
6
votes
0 answers

When is a linear map of 1-forms a pullback?

Every diffeomorphism $\phi: M\to N$ between two-dimensional compact oriented Riemannian manifolds induces a linear map on one-forms $L:\Omega^1(M)\to\Omega^1(N)$ given by the pullback of $\phi^{-1}$. Is there a simple condition for when a linear map…
user7530
  • 49,280
6
votes
0 answers

Intuition for basic concepts in differential geometry

Im studying a basic differential geometry course this semester. In the class, we defined the concept of covariant derivative (connection) as a function which takes 2 vectors fields into a vector field and satisfies some algebraic properties of…
tsufli
  • 171
6
votes
1 answer

Sub-manifold with boundary

Let $f:M\rightarrow \mathbb{R}$ be a smooth function ($M$ is a smooth manifold). Let $a$ be a regular value of $f$. Is it true that $f^{-1}(-\infty ,a]$ is a smooth manifold with boundary $f^{-1}\{a\}$? My feel is that it is correct and the proof is…
Bingo
  • 1,774
6
votes
1 answer

Local expression for a 1-form on a surface

Suppose that $\alpha$ is a non-vanishing 1-form on a 2-dimensional manifold. Why can $\alpha$ locally be written as $\alpha = f \ dg$ for some smooth functions $f$ and $g$?
Balerion_the_black
  • 2,060
6
votes
1 answer

relation between first fundamental form for different parametrization

The sphere has a parameterization map for a surface patch $\phi(u,v)=(u,v,\sqrt{1-u^2-v^2})$. It has another parametrization map for a surface patch $\beta (x,y)=(\sin x \cos y,\sin x \sin y,\cos x)$. For the first the first fundamental form comes…
abac
  • 71
5
votes
1 answer

Find the area of parallel surface

Q: Consider a surface $M$ with regular parametrization $X:U_{open}\subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and define the parallel surface $M_t$ by $$Y(u,v)=X(u,v) + tN(u,v)$$ where $N(u,v)$ is the unit normal on $X$ at point $(u,v)$ and $t$…
SamC
  • 1,738
5
votes
1 answer

Exercise about tangent space

I have just started of learning about Manifolds from Milnor's book.I am stuck on following exercise,give me some idea. Let $U$ be an open subset of a manifold $X$. Show that for any $x \in U$ the tangent space at $x$ to $U$, i.e., $T_x(U)$ is same…
Arpit Kansal
  • 10,268
5
votes
0 answers

$K = 0$ and $K = \text{const}$ surfaces produce $k_g = \text{constant}$ intersections?

Generally speaking, when do constant K (Gauss curvature) and zero K surfaces intersect to produce lines of constant geodesic curvature $ k_g $ ? Small circles on a sphere are examples. Or more specifically the question is: How should a plane be…
Narasimham
  • 40,495
5
votes
1 answer

Is $\mathbb R^n$ added by one point diffeomorphic to $S^n$?

Let $M$ be a closed smooth manifold. If for some point $p$ on $M$ we can find a diffeomorphism between $M-\{p\}$ and $\mathbb R^n$, then is $M$ diffeomorphic to $S^n$(with the standard differential structure)?
Summer
  • 6,893
5
votes
1 answer

Riemannian metric

This is a very simple question that I got confused. Is Riemannian metric a symmetric 2-tensor or symmetric 2-tensor field? Wikipedia says that it is a (0,2) tensor but my book says it is a tensor field. Are these things the same? What am I missing?…
user20353
  • 1,175
5
votes
2 answers

Basis for the set of all covariant $k$-tensors on V

Here's a proposition from Lee's Smooth Manifolds: Let $V$ be a real vector space of dimension $n$, let $(E^i)$ be any basis for $V$, and let $(\epsilon^i)$ be the dual basis. The set of all $k$-tensors of the form…
user20353
  • 1,175
5
votes
1 answer

A Problem from Docarmo's Differential Geometry

The following is a (may be simple) problem from Docarmo's Differential Geometry. Let $\alpha\colon (a,b)\rightarrow \mathbb{R}^3$ be a parametrized curve which do not pass through origin. If $\alpha(t_0)$ is a point on the trace (image) of $\alpha$…
Groups
  • 10,238
5
votes
2 answers

Curves have no intrinsic geometry

Please someone could explain me why the curves have no intrinsic geometry? With surfaces I can see that there are two kind of geometries, i.e. the euclidean one (related to euclidean isometries) and the intrinsic (related to isometries on surfaces),…
user14174
5
votes
1 answer

Background for 2 differential geometry questions

I encountered a couple of questions in a collection of differential geometry exams that I don't know how to approach. Of course I am NOT expecting a solution to these, but just a hint. If $S\subset \mathbb{R}^3$ is a non-empty surface, show that…
user54631
  • 707
1 2 3
…
99 100