An Introduction to Finsler Geometry by Xiaohuan Mo

By Xiaohuan Mo

This introductory publication 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 deal. It systematically introduces 3 periods of geometrical invariants on Finsler manifolds and their intrinsic relatives, analyzes neighborhood and worldwide effects from vintage and sleek Finsler geometry, and offers non-trivial examples of Finsler manifolds enjoyable diverse curvature stipulations.

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A • • • A dj A • • • A 6n = Pdy1 A • • • A dyi A • • • A dyn. 44) Normalizing the homogeneous coordinate (j/ 1 , • • • , yn) we have y1dy1 + •••+ yndyn = 0. 45) Chern Connection 31 6>i A • • • A 6j A • • • A 6n = f^dy1 A • • • A dyn~l +faHyl A • • • A

6 Let V be an n-dimensional vector space, and assume that F : V —> [0, +oo) is positively homogeneous of degree one and (^-)yiyi is positive definite. 38) and " — 0" only if £• and y are collinear. Proof. Setting m ••= ( ? 13) imply that ga(y)yiyj=F2(y). 37). 41) 30 An Introduction to Finsler Geometry is an inner product on V for any y e ^ \ { 0 } . 42) where equality holds iff £ and r\ are collinear. 43) where equality holds iff £ and y are collinear. &-F-1(F-1{gijyi&)2 = F- [F29iJee - to^')2] > 0 where equality holds iff £ and y are collinear.

Its second fundamental form) vanishes. It is easy to see that : M S-> (N, h) is totally geodesic if and only if carries any geodesic on M into the geodesic on N. The Landsberg spaces have the following interesting geometric characterization. 2([Ai, 1993]) Let (M,F) be a Finsler manifold and SM its projective sphere bundle. 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.

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