Download PDF by Xiaohuan Mo: An Introduction to Finsler Geometry

By Xiaohuan Mo

ISBN-10: 9812567933

ISBN-13: 9789812567932

This introductory booklet makes use of the relocating body as a device and develops Finsler geometry at the foundation of the Chern connection and the projective sphere package. It systematically introduces 3 periods of geometrical invariants on Finsler manifolds and their intrinsic family, analyzes neighborhood and worldwide effects from vintage and smooth Finsler geometry, and offers non-trivial examples of Finsler manifolds fulfilling varied curvature stipulations.

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2). Hence , Ap 2 (p 4 +3q 4 ) -2ApV (P 4 +9 4 )* _2APV (p4+q4)I y/det{gij) = J:1 + A (p 4 +? 4 )5 1 | \ _ 1 A,2(3p4+g4) I (p4+q4)t / ^JV)1 + 4 4 (p +

3). , 7, • • • < n — 1. 4 (t) dxi = ufciwfc, (3 (ii) unl = yl/F, (3 (Hi) ukzgijUij = <5fcj, (3 (iv) UkWvti =j*, (3 = 0, (3 («) (m) Fykuak ua3ui3kFFyjyk =Sap. 2 ukivjk = S/, (3 v^ = 6ijUjlglk. 1 Let UJ denote the Hilbert form of(M,F). du = 0 mod u)1 A UJ3 ; w a A cfa/^. Then Ckern Connection 27 Proof. 12). 2 • There exist ujjn £ T(T*SM), such that dcj = UJJ A ujjn. 23) where Xa/s : V ( c 5 M ) —> i? satisfy that Aa/3 = A/3a. Proof. 12), we have FyixjUv>UkllJ Au)k = F y i a . j ^ U a ' w A w ° | ^Uaj0Ja +UajU0iWa = (FxiuJ/F)ua j AW A W'3) z Awa Fy,XJ £ V " * 5 A w 3 +FyixjUa up Lj A a/ .

Then (M, F) is of Landsberg type if and only if all projective spheres in SM are totally geodesic. Similarly, set •Ja •=z Va\n = Vet where r\ = ^,ar]aea denote the Cartan form. We call Jawa Landsberg curvature and denote it by J. 3 if J = 0. 4([Shen, 1994]) A Finsler manifold is of weak Landsberg type if and only if all projective spheres in the projective sphere bundle are minimal. Let Vol(x) denote the volume of the unit Finsler sphere ([Bao and Chern, 1996]) {y£TxM\F(x,y) = l}. For functions a1 — ai{x), we have where n I dV := y/det{gij) ^ {-\)k-1ykdy1 A • • • A dyk A • • • A dyn.

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An Introduction to Finsler Geometry by Xiaohuan Mo

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