Abstract
Introduction to Geometry and geometric analysis Oliver Knill This is an introduction into Geometry and geometric analysis, taught in the fall term 1995 at Caltech. It introduces geometry on manifolds, tensor analysis, pseudo Riemannian geometry. General relativity is used as a guiding example in the last part. Exercises, midterm and nal with. In other words, Euclidean geometry is treated as geometry of initial velocities of the paths starting from a fixed point of the Riemannian space rather than the geometry of the space itself. The Riemannian construction was based on the previous study of smooth surfaces in.
For κ ⩾ 0 and r0 > 0 let ℳ(n, κ, r0) be the set of all connected, compact n-dimensional Riemannian manifolds (Mn, g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r0. We study the relation between the kth eigenvalue λk(M) of the Laplacian associated to (Mn,g), Δ = −div(grad), and the kth eigenvalue λk(X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r0) such that for all M ∈ ℳ(n, κ, r0) and X a discretization of ({M, c leqslant frac{lambda_{k}(M)}{lambda_{k}(X)} leqslant C}) for all k < |X|. Then, we obtain the same kind of result for two compact manifolds M and N ∈ ℳ(n, κ, r0) such that the Gromov–Hausdorff distance between M and N is smaller than some η > 0. We show that there exist constants c, C > 0 depending on η, n, κ and r0 such that ({c leqslant frac{lambda_{k}(M)}{lambda_{k}(N)} leqslant C}) for all ({k in mathbb{N}}).
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Correspondence to Tatiana Mantuano.
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Mathematics Subject Classification (2000): 58J50, 53C20
Supported by Swiss National Science Foundation, grant No. 20-101 469
Communicated by R. Kellerhals (Fribourg)
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Mantuano, T. Discretization of Compact Riemannian Manifolds Applied to the Spectrum of Laplacian. Ann Glob Anal Geom27, 33–46 (2005) doi:10.1007/s10455-005-5215-0
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Key words
- Laplacian
- eigenvalues
- discretization
- Hausdorff-Gromov distance
- Geometry of Geodesics and Related Topics, K. Shiohama, ed. (Tokyo: Mathematical Society of Japan, 1984), 125 - 181
Comparison and Finiteness Theorems in Riemannian Geometry
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Geometry of Geodesics and Related Topics, K. Shiohama, ed. (Tokyo: Mathematical Society of Japan, 1984), 125-181
Geometry of Geodesics and Related Topics, K. Shiohama, ed. (Tokyo: Mathematical Society of Japan, 1984), 125-181
Dates
Received: 4 January 1983
First available in Project Euclid: 3 May 2018
Received: 4 January 1983
First available in Project Euclid: 3 May 2018
Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1525309546
https://projecteuclid.org/ euclid.aspm/1525309546
Digital Object Identifier
doi:10.2969/aspm/00310125
doi:10.2969/aspm/00310125
Mathematical Reviews number (MathSciNet)
MR758652
MR758652
Zentralblatt MATH identifier
0578.53028
0578.53028
Citation
Sakai, Takashi. Comparison and Finiteness Theorems in Riemannian Geometry. Geometry of Geodesics and Related Topics, 125--181, Mathematical Society of Japan, Tokyo, Japan, 1984. doi:10.2969/aspm/00310125. https://projecteuclid.org/euclid.aspm/1525309546
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