Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces

By G. Solanes. 2005

64 citations recorded by the source corpus.

We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/420f7867dcd6a057e218bdc5c69ab57c1b9de3c8

How researchers see this paper

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In Applied Mathematics

1. A. Bernig
2. Guofang Wang
3. Chao Xia
4. Joseph H. G. Fu
5. Julian Scheuer
6. M. Makowski
7. G. Solanes
8. Jie Wu

In Astronomy and Astrophysics

1. Guofang Wang
2. Chao Xia
3. Joseph H. G. Fu
4. A. Bernig
5. Julian Scheuer
6. M. Makowski
7. Jie Wu
8. Yuxin Ge

In Geometry and Topology

1. A. Bernig
2. Joseph H. G. Fu
3. Julian Scheuer
4. M. Makowski
5. Guofang Wang
6. Chao Xia
7. G. Solanes
8. Jie Wu

Citations in the corpus

• Integral geometry of complex space forms (2012, methodology): “Proposition 3 in [Sol06] remains true, without change, in complex space forms.”
• Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere (2013, background): “In S we have the following definition of the quermassintegrals, compare [24] (also for a more detailed definition of the measure dLk): 7.”
• Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space (2013, methodology): “Curvature integrals and Quermassintegrals In this section, we recall some basic concepts in integral geometry in the hyperbolic space, we refer to Santaló’s book [35], Part IV, and Solanes’ thesis [39] for more details.”
• Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space (2018, unknown): “Recall that for a convex domain Ω in hyperbolic space, the quermassintegral Wk(Ω) is defined as follows (see [26,27]): 1 Wk(Ω) = ωk−1 · · ·ω0 ωn−1 · · ·ωn−k ∫ Lk χ(Lk ∩ Ω)dLk, k = 1, · · · , n, (1.”
• Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II (2013, background): “ ∩Ω)dL, where Lr is the space of r-dimensional totally geodesic subspaces L in Hn, ωr is the area of the r-dimensional standard round sphere and dL is the natural (invariant) measure on Lr (cf. [24], [28]). As in the Euclidean case we take W0(Ω) = Vol(Ω). With these definitions, unlike the euclidean ”