| Co-Investigator(Kenkyū-buntansha) |
IGARI Satoru Graduate School of Sciences, Tohoku University, Professor, 大学院・理学研究科, 教授 (50004289)
FUJIE Satsuro Graduate School of Sciences, Tohoku University, Lecturer, 大学院・理学研究科, 講師 (00238536)
TAKAGI Izumi Graduate School of Sciences, Tohoku University, Professor, 大学院・理学研究科, 教授 (40154744)
HORIHATA Kazuhiro Graduate School of Sciences, Tohoku University, Assistant, 大学院・理学研究科, 助手 (10229239)
NAGASAWA Takeyuki Graduate School of Sciences, Tohoku University, Associate Professor, 大学院・理学研究科, 助教授 (70202223)
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| Research Abstract |
We effectuated researches on strong lacunas of elementary solutions to the second order partial differential equations of hyperbolic type with coefficients of class CィイD1∞ィエD1, and also studies of related domains in geometry. Researches on strong labunas had made a remarkable progress by several works due to P.Gunther and his group. For a further development, however, it is indispensable to find a systematical method to construct operators whose elementary solutions admit strong lacunas. Therefore, the head investigator began by a research, from a view pint of real analysis, on normal coordinate systems which are of fundamental importance in the theory of elementary solutions.
Let n be an even or odd number not smaller than 2, and γ=(γィイD1jkィエD1)ィイD3n(/)j,k=1ィエD3 be a real symmetric non-singular matrix of order n. Take Cartesian coordinates x = (xィイD11ィエD1, …, xィイD1nィエD1) in RィイD1nィエD1. A function A(x) = (aィイD2rィエD2ィイD1sィエD1(x))ィイD3n(/)r,s=1ィエD3, with values in the space of square matri
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ces of order n, is said to be an element of a vector space V if and only if it satisfies two identities aィイD2rィエD2ィイD1bィエD1(x)γィイD2bsィエD2 = aィイD2sィエD2ィイD1bィエD1(x)γィイD2brィエD2 and xィイD1rィエD1aィイD2rィエD2ィイD1sィエD1(x) = 0. Then, there exists a one to one correspondence between an element of V and a pseudo-Riemannian metric gィイD2jkィエD2(x)dxィイD1iィエD1dxィイD1kィエD1 in a neighborhood of x = 0 having x as normal coordinates and satisfying gィイD2jkィエD2(0) = γィイD2jkィエD2. This is a theorem established by applying the theory of degenerating partial differential equations. To be more precise, we denote Y = xィイD1iィエD1ィイD7∂(/)∂xィイD1jィエD1ィエD7 and define three matrix functions S(x), N(x), R(x) involving components of basic Jacobi fields, of the Levi-Civita connection and of the curvature tensor, respectively. Then, they satisfy three partial differential equations YS = -SN, YN+N = NィイD12ィエD1+R, YYS+YS = -SR. At the origin, S = I (the unit matrix), N = 0 and R = 0. Given one of S, N, R, we can solve partial differential equations written above to find two others in one and only one way. In particular, we have S = (σィイD2jィエD2ィイD1BィエD1), from which we obtain a metric tensor gィイD2jkィエD2 via the equality gィイD2jkィエD2 = σィイD2jィエD2ィイD1AィエD1γィイD2ABィエD2σィイD2kィエD2ィイD1BィエD1 in a neighborhood of x = 0. In this way, we can define a metric from any element of V because N and R belong to V, and vice versa. Less
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