Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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### About the limit of transverse intersection

Let $n$ be a fixed positive integer, and let $W^{s}(R_{q})$ and $W^{u}(R_{q})$ be the stable and unstable manifolds of a fixed point $R_{q}$ of a discrete 2-D mapping. Notice that the sequence $R_{q}$ ...

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45 views

### Does an 8 dimensional compact Riemannian manifold contain an embedded minimal hypersurface?

It is well know that if $(M^{n+1},g)$ is a compact Riemannian manifold and $n \leq 6$ then there exists a smooth, embedded minimal hypersurface $\Sigma^n$ in $M$ (infinitely many such $\Sigma$, even)
...

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98 views

### Problem about the homology groups of the complex projective space $\mathbb{C}P_n$ [on hold]

My question is how we can compute the homology groups of the complex projective space $\mathbb{C}P_n$ by the following Corollary5.4 at page 31 in Milnor's book：
Corollary5.4
If $c_{\lambda+1}=c_{\...

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162 views

### Physical (GR) Differential Geometry?

I apologize for the vague question, but are there instances, within the theme of "Physical Mathematics" ($=$ physically motivated mathematics) of some calculation from General Relativity shedding ...

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102 views

### A geometric rank of Riemannian manifolds

There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks:
The maximum number of global independent vector fields which can be defined ...

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81 views

### Holomorphic map, Instantons of Complex Projective Space and Loop Group

It seems that holomorphic (or rational) maps play a crucial role to relate the following data:
Instanton in 1-dimensional complex Projective Space $$\mathbb{P}^1$$
in a 2 dimensional (2d) spacetime.
...

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104 views

### Convexity of curves in Riemannian surfaces

It is known that a curve $f:[0,2\pi]\to \mathbf{R}^2$ is convex if $\partial_t (\arg f'(t))\ge 0$. My question is: does this statement have an analogue in the setting of Riemannian surfaces instead of ...

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72 views

### Is it necessary for a conservative vector field to be $C^1$?

Is it necessary for a conservative vector field on a domain $A \subset \mathbb{R}^3$ to be $C^1$?
I know that if a vector field $F$ is defined on a simply connected domain $A$ simply connected and $...

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59 views

### geodesic balls in the conformal change

Consider a compact smooth Riemannian manifold $(\mathcal{M}, g)$. We consider a conformal change of the metric. Let $\tilde{g}=\phi g$, where $\phi$ is smooth and positive. Moreover, it satisfies $...

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94 views

### Polygon of convex arcs

Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists).
Assume ...

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99 views

### Minkowski isometries

Consider theorem 1.7 from chapter III of 'Elementary differential geometry' by O'Neill. It says that:
Theorem 1.7: If $\phi$ is an isometry of $E^3 $, then there exists a unique translation $T$ and a ...

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81 views

### Sectional curvatures under simple maps

Suppose that we have a submanifold $X$ of $\mathbb{R}^n$ with the induced Euclidean metric, whose sectional curvatures we have a handle of (say, they are lower bounded by some $\kappa$).
Is there a ...

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74 views

### Foliation with a compact leaf

Let $M$ be a closed oriented manifold, and $F$ be a foliation of $M$. We assume the dimension and codimension of $F$ are both greater than $1$.
Q Under what condition, we can say that $F$ admits one ...

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24 views

### Decomposition of a Jacobi field along a lightlike geodesic

Consider a Lorentzian manifold of dimension $1+n$ (with $n\geq1$) and a lightlike geodesic $\gamma(t)$ on it.
One can define a Jacobi field $J(t)$ along $\gamma$ in the usual way without issues.
In ...

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208 views

### When $\int_M \exp(-d_M(x,y)^2/t) dvol(y)$ becomes constant for a Riemannian manifold $M$?

Let $(M,g)$ be a closed and connected Riemannian manifold. $d_M$ is its geodesic metric and $dvol_M$ is its standard volume measure. For each $t>0$, define a map $f:M\rightarrow\mathbb{R}_{>0}$ ...