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Register a new userCharacteristic Points, Fundamental Cubic Form and Euler Characteristic of Projective Surfaces
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We define local indices for projective umbilics and godrons (also called cusps of Gauss) on generic smooth surfaces in projective 3-space. By means of these indices, we provide formulas that relate the algebraic numbers of those characteristic points on a surface (and on domains of the surface) with the Euler characteristic of that surface (resp. of those domains). These relations determine the possible coexistences of projective umbilics and godrons on the surface. Our study is based on a fundamental cubic form for which we provide a closed simple expression.
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A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study vario
Baez asks whether the Euler characteristic (defined for spaces with finite homology) can be reconciled with the homotopy cardinality (defined for spaces with finite homotopy). We consider the smallest infinity category $text{Top}^text{rx}$ containing both these classes of spaces and closed under homotopy pushout squares. In our main result, we compute the K-theory $K_0(text{Top}^text{rx})$, which is freely generated by equivalence classes of connected p-finite spaces, as p ranges over all primes. This provides a negative answer to Baezs question globally, but a positive answer when we restrict attention to a prime.
The Euler characteristic $chi =|V|-|E|$ and the total length $mathcal{L}$ are the most important topological and geometrical characteristics of a metric graph. Here, $|V|$ and $|E|$ denote the number of vertices and edges of a graph. The Euler characteristic determines the number $beta$ of independent cycles in a graph while the total length determines the asymptotic behavior of the energy eigenvalues via the Weyls law. We show theoretically and confirm experimentally that the Euler characteristic can be determined (heard) from a finite sequence of the lowest eigenenergies $lambda_1, ldots, lambda_N$ of a simple quantum graph, without any need to inspect the system visually. In the experiment quantum graphs are simulated by microwave networks. We demonstrate that the sequence of the lowest resonances of microwave networks with $beta leq 3$ can be directly used in determining whether a network is planar, i.e., can be embedded in the plane. Moreover, we show that the measured Euler characteristic $chi$ can be used as a sensitive revealer of the fully connected graphs.
We revisit Allendoerfer-Weils formula for the Euler characteristic of embedded hypersurfaces in constant sectional curvature manifolds, first taking some time to re-prove it while demonstrating techniques of [2] and then applying it to gain new understanding of isoparametric hypersurfaces.
We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the main known examples of bundles with non-zero signature.
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