Positive sectional curvatures does not imply positive Gauss-Bonnet integrand

Superintelligence Does Not Imply Benevolence

Asmachines become capable ofmore autonomous and intelligent behavior, will they also display more morally desirable behavior? Earth’s history tends to suggest that increasing intelligence, knowledge, and rationality will result in more cooperative and benevolent behavior. Animals with sophisticated nervous systems track and punish exploitative behavior, while rewarding cooperation. Humans form ...

The Gauss-Bonnet Theorem

The Gauss-bonnet Theorem

The Gauss Bonnet theorem links differential geometry with topology. The following expository piece presents a proof of this theorem, building up all of the necessary topological tools. Important applications of this theorem are discussed.

Gauss - Bonnet brane cosmology

We consider 5-dimensional spacetimes of constant 3-dimensional spatial curvature in the presence of a bulk cosmological constant. We find the general solution of such a configuration in the presence of a Gauss-Bonnet term. Two classes of non-trivial bulk solutions are found. The first class is valid only under a fine tuning relation between the Gauss-Bonnet coupling constant and the cosmologica...

Visualization of Gauss-Bonnet Theorem

The sum of external angles of a polygon is always constant, π 2 . There are several elementary proofs of this fact. In the similar way, there is an invariant in polyhedron that is π 4 . To see this, let us consider a regular tetrahedron as an example. Tetrahedron has four vertices. Three regular triangles gather at each vertex. Developing the tetrahedron around each vertex, there is an open ang...