Differential Geometry Lecture Notes by Christian Bär

Christian B¨ ar
Differential Geometry
Summer Term 2013
Version of August 26, 2013
GEOMETRY IN POTSDAM

The titlepage was created using 3D-XplorMath and gimp.

Contents
Preface iii
1 Manifolds 1
1.1 Topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Tangent vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Directional derivatives and derivations . . . . . . . . . . . . . . . . . . . 23
1.5 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Semi-Riemannian Geometry 35
2.1 Bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Semi-Riemannian metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Differentiation of vector fields . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Vector fields along maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.5 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.6 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3 Curvature 79
3.1 The Riemannian curvature tensor . . . . . . . . . . . . . . . . . . . . . . . 79
3.2 Sectional curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3 Ricci- and scalar curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.4 Jacobi fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4 Submanifolds 107
4.1 Submanifold of differentiable manifolds . . . . . . . . . . . . . . . . . . . 107
4.2 Semi-Riemannian submanifolds . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3 Totally geodesic submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.5 Trigonometry in spaces of constant curvature . . . . . . . . . . . . . . . . 133
5 Riemannian Geometry 143
5.1 The Riemannian distance function . . . . . . . . . . . . . . . . . . . . . . 143
5.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.3 The second variation of the energy . . . . . . . . . . . . . . . . . . . . . . 156
5.4 The Bonnet-Myers theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Literature 163
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Contents
Index 165
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Preface
These are the lecture notes of an introductory course on differential geometry that I
gave in 2013. It introduces the mathematical concepts necessary to describe and ana-
lyze curved spaces of arbitrary dimension. Important concepts are manifolds, vector
fields, semi-Riemannian metrics, curvature, geodesics, Jacobi fields and much more.
The focus is on Riemannian geometry but, as we move along, we also treat more gen-
eral semi-Riemannian geometry such as Lorentzian geometry which is central for ap-
plications in General Relativity. We also make a connection to classical geometry when
we apply differential geometry to derive the laws of trigonometry on spaces of constant
curvature. One fundamental result of Riemannian geometry that we show towards the
end of the course is the Bonnet-Myers theorem. It roughly states that the larger the
curvature of a space, the smaller the space itself must be.
The lecture course did not require prior attendance of a course on elementary differ-
ential geometry treating curves and surfaces but such a course would certainly help to
develop the right intuition.
It is my pleasure to thank all those who helped to improve the manuscript by sugges-
tions, corrections or by work on the L
A
T
E
X code. My particular thanks go to Andrea
R¨oser who wrote the first version in German language and created many pictures in
wonderful quality, to Volker Branding who translated the manuscript into English and
to Ramona Ziese who improved the layout.
Potsdam, August 2013
Christian B¨ar
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