By David Dai-Wai Bao

In Riemannian geometry, measurements are made with either yardsticks and protractors. those instruments are represented through a relatives of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in precept outfitted with just a family members of Minkowski norms. So yardsticks are assigned yet protractors usually are not. With any such restricted device package, it's typical to ask yourself simply how a lot geometry you can still discover and describe? It now seems to be that there's a average solution. Finsler geometry features a reliable repertoire of pressure and comparability theorems, so much of them based upon a fruitful analogue of the sectional curvature. there's additionally a bewildering array of particular examples, illustrating many phenomena which admit merely Finslerian interpretations. This publication makes a speciality of the uncomplicated yet crucial goods between those effects. a lot concept has long past into making the account a teachable one. David Bao is Professor of arithmetic and of the Honors collage, on the college of Houston. He acquired his Ph. D. from the college of California at Berkeley in 1983, with Jerry Marsden as his consultant. prior to coming to Houston, he did years of post-doctoral reviews on the Institute for complicated learn in Princeton, New Jersey. along with differential geometry, he's passionately inquisitive about the methods cats and goldfish imagine. Shiing-Shen Chern is Professor Emeritus of arithmetic on the college of California at Berkeley, and Director Emeritus of the Mathematical Sciences learn Institute. he's additionally special traveling Professor Emeritus on the collage of Houston. Chern bought his D. Sc. in 1936, as a scholar of W. Blaschke. He conducted his post-doctoral stories lower than E. Cartan. Chern has garnered plenty of differences up to now. those contain the Chauvenet Prize (1970), nationwide Medal of technological know-how (1975), the Humboldt Award (1982), the Steele Prize (1983), and the Wolf origin Prize (1983-84). Zhongmin Shen is affiliate Professor of arithmetic at Indiana collage Purdue college Indianapolis (IUPUI). He earned his Ph. D. from the country college of recent York at Stony Brook in 1990 below Detlef Gromoll. He spent 1990-91 on the Mathematical Sciences learn Institute at Berkeley, and 1991-93 as a Hildebrandt Assistant Professor on the collage of Michigan at Ann Arbor.

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**Example text**

They are totally umbilic. 5. Corollary ([HOD. I f S 2 is immersed in R 3 with non-zero constant mean curvature H , then S 2 is a standard sphere embedded in R 3. 6. Corollary ([AI],[Cb]). 7. Corollary. I f S 2 is immersed in S 3 with non-zero constant mean curvature H , then S 2 is a standard sphere, which is the intersection o f S 3 and an affine hyperplane o f R 4. Next we discuss the immersions of closed surfaces with genus greater than zero in Na(c). 10) 48 Part I Submanifold Theory hi1 e2u, hi1 : b12 - - h12 e2u.

Exercises. 1. Find all isometries of (R '~, 9), where g is the standard metric. 2. If ~ is a Killing vector field and v a smooth tangent vector field on M , then (Vv~,V) = 0. 3. Let X be a smooth Killing vector field on the closed Riemannian manifold M . Show that (i) ~A(IIXII ~) = - mc(X, x ) + IlVXll ~. , Ric(X, X) _< 0 for all vector field X) then the dimension of the group of isometries of M is O. Chapter 2. Local Geometry of Submanifolds. Given an immersed submanifold M '~ of the simply connected space form Nn+k(c) there are three basic local invariants associated to M: the first and second fundamental forms and the normal connection.

A en+l i,o~>n+ l + E el c~>n+l A . . A e n A ~n+l,o~eoz : 0. Hence the span of e l ( X ) , . . , en+l(X) is a fixed (n + 1)-dimensional linear subspace V o f R '~+k for all x E M. But X = xo-e,~+a/a, so M is contained in the intersection of the affine (n + 1)-plane Xo + V and the hypersphere of R n+k of center Xo and radius 1/a. • Exercises. 1. 9 for totally umbilic hypersurfaces of S n+l and I n+l. 2. 11 for totally umbilic submanifolds of S '~+1 and H n+l. 3 Fundamental theorem for submanifolds of space forms.