Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture
Regular price
€229.40
14 days return policy
Shipping & Delivery
A01=JeongHyeong Park
A01=John V. Leahy
A01=Peter B. Gilkey
Author_JeongHyeong Park
Author_John V. Leahy
Author_Peter B. Gilkey
Bochner Laplacian
Category=PBMP
compact manifolds
differential topology
eigenvalue analysis
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Laplace type operators
mathematical physics and relativity
Neumann boundary conditions
principal fiber bundles
rigidity theorems
spectral analysis of Laplacians on manifolds
spectral geometry
Product details
- ISBN 9780849382772
- Weight: 594g
- Dimensions: 156 x 234mm
- Publication Date: 27 Jul 1999
- Publisher: Taylor & Francis Ltd
- Publication City/Country: GB
- Product Form: Hardback
Secure
checkout
Fast Shipping
Easy returns
This cutting-edge, standard-setting text explores the spectral geometry of Riemannian submersions. Working for the most part with the form valued Laplacian in the class of smooth compact manifolds without boundary, the authors study the relationship-if any-between the spectrum of Dp on Y and Dp on Z, given that Dp is the p form valued Laplacian and pi: Z ® Y is a Riemannian submersion.
After providing the necessary background, including basic differential geometry and a discussion of Laplace type operators, the authors address rigidity theorems. They establish conditions that ensure that the pull back of every eigenform on Y is an eigenform on Z so the eigenvalues do not change, then show that if a single eigensection is preserved, the eigenvalues do not change for the scalar or Bochner Laplacians. For the form valued Laplacian, they show that if an eigenform is preserved, then the corresponding eigenvalue can only increase. They generalize these results to the complex setting as well. However, the spinor setting is quite different. For a manifold with non-trivial boundary and imposed Neumann boundary conditions, the result is surprising-the eigenvalues can change.
Although this is a relatively rare phenomenon, the authors give examples-a circle bundle or, more generally, a principal bundle with structure group G where the first cohomology group H1(G;R) is non trivial. They show similar results in the complex setting, show that eigenvalues can decrease in the spinor setting, and offer a list of unsolved problems in this area.
Moving to some related topics involving questions of positive curvature, for the first time in mathematical literature the authors establish a link between the spectral geometry of Riemannian submersions and the Gromov-Lawson conjecture.
Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture addresses a hot research area and promises to set a standard for the field. Researchers and applied mathematicians interested in mathematical physics and relativity will find this work both fascinating and important.
Peter B. Gilkey, John V. Leahy, both University of Oregon, Eugene, USA. JeongHyeong Park, Sungkyunkwan University.
