Geodesically equivalent metrics on homogenous spaces

summary:Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two $G$-invariant metrics of arbitrary signature on homogenous space $G/H$ are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that the existence of nonproportional, geodesically equivalent, $G$-invariant metrics on homogenous space $G/H$ implies that their holonomy algebra cannot be full. We give an algorithm for finding all left invariant metrics geodesically equivalent to a given left invariant metric on a Lie group. Using that algorithm we prove that no two left invariant metrics of any signature on sphere $S^3$ are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, nonproportional, left-invariant metrics

On modeling of competencies in a descriptive geometry course

As we know the reform of higher education through last 15 years has been done in such a way that system is changed from teaching to educational system. It involves: a shift from the lecture form to a student-centered approach, a shift in the paradigm from measuring educational attainment to measuring of competence. Problem solving and critical way of thinking are among the most important competencies in 21st Century by the most expert’s opinion in this area.Our main goal is to present how one can model these competencies of students of Mathematics and Computer Science in the frame of Descriptive Geometry course. We point out the role of computers and software to reach these competencies in Descriptive Geometry course, having in mind that our students have developed abstract way of thinking in previous courses. We use a questionnaire-based approach to conclude which method best fulfils the previously mentioned competencies

Self-duality and pointwise Osserman manifolds

summary:This paper is a contribution to the mathematical modelling of the hump effect. We present a mathematical study (existence, homogenization) of a Hamilton-Jacobi problem which represents the propagation of a front f$$lame in a striated media

Applications of Spectral Geometry to Affine and Projective Geometry

We use the asymptotics of the heat equation to construct spectral invariants in affine and projective geometry

When the Leading Terms in the Heat Equation Asymptotics Are Coercive

: Let M be a compact Riemannian manifold without boundary of dimension m 2: Let an (D) be the asymptotics of the heat equation for n 3: If D is the p form valued Laplacian for 0 p m; the an lead to coercive estimates for the highest order jets of the covariant derivatives of the curvature tensor for any n: If D is the conformal Laplacian, the an lead to coercive estimates if and only if 2n = 2 fm \Gamma 2; m \Gamma 1; mg: If D is the spinor Laplacian, the an lead to coercive estimates if and only if 2n ? m: Classification number: 58G25. Let M be a compact manifold of dimension m without boundary. Let \Delta p := \Delta p (g) = (d p\Gamma1 ffi p\Gamma1 + ffi p d p ) (1) be the Laplacian on p forms determined by a Riemannian metric g: We say that two Riemannian metrics g and ~ g are p \Gamma isospectral if \Delta p (g) and \Delta p (~g) have the same spectrum. Let L be the conformal Laplacian and let \Delta s be the spin Laplacian; the terms L \Gamma isospectral and spin-isospectra..