Books

by Zhongmin Shen

10/13/11

 

 

 

 

           

Table of Content

1 Randers Spaces 1

1.1 Randers Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Distortion and Volume Form . . . . . . . . . . . . . . . . . . . . 5

1.3 Cartan Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Randers Metrics and Geodesics 15

2.1 Randers Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Zermelo’s Navigation Problem . . . . . . . . . . . . . . . . . . . . 18

2.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Randers Metrics of Berwald type . . . . . . . . . . . . . . . . . . 25

3 Randers Metrics of Isotropic S-curvature 29

3.1 S-curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Isotropic S-curvature in terms of  alpha and  beta. . . . . . . . . . . . . 31

3.3 Isotropic S-curvature in terms of h and W . . . . . . . . . . . . . 35

3.4 Examples of Isotropic S-curvature . . . . . . . . . . . . . . . . . 39

3.5 Randers Metrics with Secondary Isotropic S-Curvature . . . . . . 45

4 Riemann Curvature and Ricci Curvature 53

4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Riemann Curvature of Randers Metrics . . . . . . . . . . . . . . 55

4.3 Randers Metrics of Scalar Flag Curvature . . . . . . . . . . . . . 57

5 Projective Geometry of Randers Spaces 65

5.1 Projective Quantities . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Douglas-Randers Metrics . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Weyl-Randers Metrics . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4 Generalized Randers DW-metrics . . . . . . . . . . . . . . . . . . 73

6 Randers Metrics with Special Riemann Curvature Properties 81

6.1 Ricci-quadratic Randers metrics . . . . . . . . . . . . . . . . . . . 81

6.2 Randers Metrics of R-quadratic Curvature . . . . . . . . . . . . . 82

6.3 Randers metrics of W-quadratic curvature . . . . . . . . . . . . . 84

6.4 Randers Metrics of Sectional Flag Curvature . . . . . . . . . . . 86

7 Randers Metrics of Weakly Isotropic Flag Curvature 95

7.1 Weak Einstein Randers Metrics . . . . . . . . . . . . . . . . . . . 95

7.2 Randers Metrics of Weakly Isotropic Flag Curvature . . . . . . . 100

7.3 Solutions via Navigation . . . . . . . . . . . . . . . . . . . . . . . 105

7.4 Weak Einstein Randers Metrics via Navigation Data . . . . . . . 111

8 Projectively Flat Randers Metrics 117

8.1 Projectively Flat Randers Metrics of Constant Flag Curvature . 117

8.2 Projectively Flat Randers Metrics of Weakly Isotropic Flag Curvature . . . . . . . .120

8.3 Projectively Flat Randers Metrics on S^n    ............................................................... 127

9 Conformal Geometry of Randers Metrics 133

9.1 Conformally Invariant Spray . . . . . . . . . . . . . . . . . . . . . 133

9.2 Conformally Flat Randers Metrics . . . . . . . . . . . . . . . . . 137

9.3 Conformally Berwaldian Randers Metrics . . . . . . . . . . . . . 139

10 Dually Flat Randers Metrics 143

10.1 Dually Flat Finsler Metrics . . . . . . . . . . . . . . . . . . . . . 143

10.2 Dually Flat Randers Metrics . . . . . . . . . . . . . . . . . . . . 146

10.3 Dually Flat Randers Metrics with Isotropic S-curvature . . . . . 149

Index 154