Tagged Questions

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 ...

134 views

Books on differential geometry in the cases $n=2$ and $n=3$

I'm interested in learning the differential geometry of standard, "physical" space, that is $\mathbb R^2$ and $\mathbb R^3$. The sort of problems that were studied in the 18th and 19th century... ...
35 views

54 views

584 views

Topology of biological compartments

In the field of cell biology, there is a general sub-field concerned with the topology of organelle membranes, and a key focus remains on how these dynamic membranes deform and interact with cellular ...
73 views

Twisting with a degree negative line bundle

Let $X$ be a Riemann surface. Let $M_1$ and $M_2$ be two holomorphic bundles on $X$. Does injectivity of $h^0(M_1)\to h^0(M_2)$ imply $h^0(M_1\otimes L)\to h^0(M_2\otimes L)$ is injective? Where $L$ ...
102 views

70 views

Exercise on left invariant 2-forms

I'm trying to solve a problem about the Lie group of transformation over $\mathbb{R}$: $$x\mapsto ax+b,$$ where $a,b\in\mathbb{R}, a\neq 0.$ I'm asked to find the space of ...
62 views

Pseudo-scalar product on Manifold

I'm trying to study the Semi-Riemannian Manifold and the relativity (I use the book Semi-Riemannian Manifold- O'Neill). But I don't understand the following thing: In a Semi-Riemannnian Manifold, I ...
85 views

Do Anosov flows exist on two dimensional compact manifolds?

Question: Let $M$ denote a two-dimensional, smooth, compact Riemannian manifold (without boundary). If $\phi:\mathbb{R}\times M \rightarrow M$ is a smooth flow, then prove that $\phi$ is not an ...
53 views

Ribbon Surfaces and Legendrian Graphs on Contact 3-manifolds.

Let $M=(M, \xi)$ be a contact 3-manifold. I am trying to show that every Legendrian graph L (i.e., a graph embedded in $M$ so that it is everywhere-tangent to the contact planes) admits a ribbon ...
52 views

Sign of first chern class with some conditions

Let $X$ be a compact Kahler algebraic variety which $K_M$ is big and nef, and $Kod(X)=dimX$ then why the first chern class $c_1(M)$ is negative or zero . I don't undrestand kawamata's theorem in this ...
Let $G$ be a Lie group and $\iota \colon G\to G$ denote the inversion. If $e$ is the identity of $G$, prove that: $$\text d \iota _e = -\text {id} _{T_e G}.$$ I understand that the differential at $e$ ...
2-forms on $S^2$
I've read that the group $H^2_{dR}(S^2)=\mathbb{R}$. If I'm not wrong, this implies that one can build closed 2-forms that are not exact. Can somebody show me an example, please? Thanks!