Commuting Operators and Separation of Variables for Laplacians of Projectively Equivalent Metrics |
| |
Authors: | Matveev Vladimir S |
| |
Institution: | (1) Isaac Newton Institute, Cambridge, CB3 0EH, U.K. |
| |
Abstract: | Let two Riemannian metrics g and g on one manifold M
n
have the same geodesics (considered as unparameterized curves). Then we can construct invariantly n commuting differential operators of second order. The Laplacian
g
of the metric g is one of these operators. For any x M
n
, consider the linear transformation G of T
x
M
n
given by the tensor g
I g j
. If all eigenvalues of G are different at one point of the manifold then they are different at almost every point; the operators are linearly independent and their symbols are functionally independent. If all eigenvalues of G are different at each point of a closed manifold then it can be covered by the n-torus and we can globally separate the variables in the equation
g
f = f on this torus. |
| |
Keywords: | integrable systems quantum integrable systems commutative operators projectively equivalent metrics geodesically equivalent metrics separation of variables Levi-Civita connection Levi-Civita coordinates |
本文献已被 SpringerLink 等数据库收录! |