Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 287evolution equation for scalar curvature R, which is∂ t R = ∆R + 2 |R ij | 2 .In dimension 3, positive Ricci curvature is preserved under the Ricci flow.This is a special feature of dimension 3 and is related to the fact that theRiemann curvature tensor may be recovered algebraically from the Riccitensor and the metric in dimension 3. Positivity of sectional curvatureis not preserved in general. However, the stronger condition of positivecurvature operator is preserved under the Ricci flow.3.10.2.4 Structure Equations on MLet {X a } m a=1, {Y i } n i=1 be local orthonormal framings on M, N respectivelyand {e i } n i=1 be the induced framing on E defined by e i = Y i ◦ φ, then thereexist smooth local coframings {ω a } m a=1, {η i } n i=1 and {φ∗ η i } n i=1 on T M, T Nand E respectively such that (locally)g =m∑ω 2 a and h =a=1n∑η 2 i .The corresponding first structure equations are [Mustafa (1999)]:i=1dω a = ω b ∧ ω ba , ω ab = −ω ba ,dη i = η j ∧ η ji , η ij = −η ji ,d(φ ∗ η i ) = φ ∗ η j ∧ φ ∗ η ji , φ ∗ η ij = −φ ∗ η ji ,where the unique 1–forms ω ab , η ij , φ ∗ η ijforms. The second structure equations areare the respective connectiondω ab = ω ac ∧ ω cb + Ω M ab, dη ij = η ik ∧ η kj + Ω N ij ,d(φ ∗ η ij ) = φ ∗ η ik ∧ φ ∗ η kj + φ ∗ Ω N ij ,where the curvature 2–forms are given byΩ M ab = − 1 2 RM abcdω c ∧ ω d and Ω N ij = − 1 2 RN ijklη k ∧ η l .The pull back map φ ∗ and the push forward map φ ∗ can be written as[Mustafa (1999)]φ ∗ η i = f ia ω a