0
votes
1answer
267 views

Yang-Mills equations are not elliptic [closed]

How does one prove that the Yang-Mills equations (from classical Yang-Mills theory) are not elliptic? Alternatively, how does one calculate the principal symbol of the Yang-Mills equations? Can ...
-4
votes
1answer
122 views

compact complex manifolds and complet curves [closed]

let $X$ be a compact complex manifold of dimension one . my first question is : 1) -does all compact complex manifolds of dimension one admit nonconstant meromorphic function ? . now , let ...
6
votes
2answers
379 views

The trace of a wedge product of matrices

I'm trying understand a computation on page 371 of Besse's book on Einstein Manifolds. I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form ...
1
vote
1answer
128 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
0
votes
1answer
184 views

Non-Symmetric Equivariant Riemannian Metrics on Homogeneous Spaces

For a homogeneous space $M = G/H$, the number of $H$-equivariant Riemannian metrics on $M$ is usually much smaller than the space of Riemannian metrics. I am wondering what happens when the symmetric ...
3
votes
2answers
237 views

Real analytic submanifolds of $\mathbb{R}^{n}$

Hallo, Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as ...
5
votes
2answers
1k views

Metric associated to a Connection on a Vector Bundle

General question: Given a vector bundle $E \rightarrow M$ on a complex manifold $M$, and a connection $\nabla$ on $E$, is it possible to find an Hermitian structure on $E$ such that $\nabla$ is the ...
8
votes
4answers
819 views

Riemannian metric on a flag variety

$\def\C{\mathbb{C}}\def\CP{\mathbb{CP}}$Every complex projective space $\CP^n$ has a natural Riemannian metric, the Fubini–Study metric, which is defined via the quotient definition of $\CP^n = ...
4
votes
2answers
821 views

Invariant Metrics on the Sphere

I've been thinking about $SU(n)$-invariant metrics on the odd-dimensional spheres $S^{2n-1} \simeq SU(n)/SU(n-1)$. For $S^1$, all such metrics are in correspondence with the positive reals. For $S^{3} ...
1
vote
2answers
342 views

Holonomy group of $\mathbb{O}P^1$

What is the holonomy group of the 1-dimensional octonionic projective space ?
3
votes
2answers
403 views

riemannian length of an element of the fundamental group of a manifold

It is a stupid question i guess but like they say if you ask you are stupid for 5 minutes and if you don't ask you are stupid forever . here is the question given a closed manifold $(M,g)$ and ...
7
votes
2answers
937 views

Kähler metrics for projective space that are not the Fubini-Study metric

For projective $N$-space $CP^{N}$, there is a canonical Kähler metric called the Fubini-Study metric. Do there exist other Kähler metrics for $CP^N$. If so, is there any classification of such ...
13
votes
4answers
985 views

Algebraic surfaces and their (intrinsic) geometry

Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and ...
6
votes
1answer
888 views

Is the cotangent bundle to a Kahler manifold hyperkahler?

Let me be more specific. Let $M$ be a Kahler manifold with Riemannian metric $g$ and complex structure $I$. Then $T^\ast M$ will also be Kahler with metric and complex structure induced from $M$ (I ...
4
votes
2answers
271 views

volume growth of tubular neigbhorhood of critical values of an algebraic/differentiable map

Edit: One can also assume $M,N$ compact in the following, and that $M$ is equipped with a Riemannian metric as well. Incorporated Mike's suggestion. Sard's theorem says that for every smooth map $f: ...
9
votes
3answers
2k views

Calabi - Yau Manifolds

I just started reading about Calabi-Yau manifolds and most of the sources I came across defined Calabi-Yau manifold in a different way. I can see that some of them are just same and I can derive one ...
10
votes
4answers
943 views

Hermitian symmetric spaces vs Hermitian homogeneous spaces

A Hermitian symmetric space is a connected complex manifold with a hermitian metric on which the group of holomorphic isometries acts transitively, and which satisfies the following extra condition: ...
17
votes
3answers
879 views

Algebraic (semi-) Riemannian geometry ?

I hope these are not to vague questions for MO. Is there an analog of the concept of a Riemannian metric, in algebraic geometry? Of course, transporting things literally from the differential ...
30
votes
5answers
2k views

Intuition behind moduli space of curves

For a genus g compact smooth surface $M$, an algebraic structure is the same as a complex structure is the same as a conformal structure. So the moduli space of smooth curves should be the same as the ...