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6
votes
0answers
219 views

Negative curvature in the middle of $R^{3}$

What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside? Basically, I am asking for a ...
3
votes
0answers
139 views

Strake's splitting theorem for complex sectional curvature

Strake's splitting theorem: Let $(M^n,g)$ be an open manifold with sectional curvature $K\ge 0$ and soul $\Sigma^k$. If the normal holonomy group of $\Sigma$ is trivial, $M$ splits isometrically into ...
2
votes
0answers
118 views

Manifold with a quasi-positive curvature

As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a quasi-positive curvature to be a manifold with positive curvature. But ...
1
vote
0answers
83 views

Exposition of the Calabi complex

I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature". The complex is referenced as "Calabi complex" in various citing ...
1
vote
0answers
219 views

Formula for the curvature of an induced connection

Let $\pi_P:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connection $A$ and $\pi_Q:Q\to M$ be a principal $H$-bundle on a manifold $M$. Let $f:P\to Q$ be a morphism of bundles ...
0
votes
0answers
195 views

Integral of Square of Mean Curvature

Let us assume $\text{H}$ is the mean curvature of a compact surface in $\mathbb E^3$ and $g$ is its genus. When $g$ is arbitrary, we have $\int_{\mathbb S^2}\text{H}^2dV=4\pi$ and ...
0
votes
0answers
173 views

Value of convolution integral of Gaussian function and curvature of a circle segment

Is there an analytical expression for the following integral, assuming $\alpha\in(0,\pi)$ and $\sigma>0$? $$-\frac{1}{\sqrt{2\pi} \sigma} \int_{-\sin(\alpha/2)}^{\sin(\alpha/2)} \exp(-\tfrac12 ...
0
votes
0answers
115 views

General form of a metric affine connection with zero curvature

I have read somewhere that the general form of a metric affine connection whose Riemann curvature is zero is given by $$\Gamma^{i}_{ij}=A^{i}_{\alpha}\partial_{j}A^{\alpha}_{k}$$ where ...