 -topology 6
 -manifold structure of  19
-manifold structure on  19
 -respecting connection 100
 -respecting connection 92
 -respecting connection 98
Algebraic bracket 28
Algebraic derivation 27
Anti self dual 99
Approximation property (bornological) 15
Associated bundle 48
Associated vector bundles 52
Base space 36
Basis 36
Bianchi identity 37
Bundle 36
Bundle atlas 36
Bundle of gauges 54
Cartesian closedness 8 11 13
Characteristic class 94
Chern class 97
Christoffel forms 39
Classifying connection 87
Classifying space 53 87 89
Cocurvature 33
Cocycle condition 36
Cocycle of transition functions 36
Cohomologous 44
Complete connection 41
Connection 32 37 92
Connection form 95
Connection form (Lie algebra valued) 60
Connector 69
Continuous derivation over eva 14
Convenient vector space 7
Covariant derivative 63 70
Covariant derivative on vector bundles 70
Covariant exterior derivative 63 71
Curvature 32 33 37 70
Curvature form 96
Curvature form (Lie algebra valued) 60
De Rham cohomology of M with twisted coefficient domain  92
Derivations (graded) 27
Diffeomorphism group 22
Ehresmann connection 41
Equivariant mappings and associated bundles 50
Exponential law 20
f-dependent 33
F-related 33
Fiber bundle 36
Fiber bundle atlas 36
Fiber chart 36
Fiber volume 90
Fiberwise symplectic form 98
Flat connections 38
Frame bundle 81
Frolicher — Nijenhuis bracket 29
Fundamental vector field 5
G-atlas 44
G-bundle 44
G-bundle structure 44
Gauge group of a fiber bundle 83
Gauge transformations 54
General Bianchi identity 33
Generalization of the theorem of Ambrose and Singer 77
Global formula for the Frolicher — Nijenhuis bracket 30
Grassmann manifold 46
Hodge-*-operator 99
Holomorphic 10
Holomorphic curves 10
Holonomy group 65 76
Holonomy groups for principal connections 65
Holonomy Lie algebra 76
Homogeneous spaces 45
Homomorphism of G-bundles 51
Homomorphism over  of principal bundles 47
| Horizontal bundle 37
Horizontal differential forms 63
Horizontal foliation 38
Horizontal lift 37
Horizontal projection 37
Horizontal space 32
Horizontal vectors 37
Induced connection 66 61
Inducing principal connections on associated vector bundles 69
Irreducible principle connection 66
Kinematic tangent vector 14
Lie derivation 28
Lie group 5
Linear connection 69
Linear frame bundle 52
Local description of connections 38
Local descriptions of principal connections 62
Local formulas for the Frolicher — Nijenhuis bracket 31
Manifold structure of 17
Maurer — Cartan formula 39
n-transitive 23
Natural bilinear concomitants 34
Naturality of the Frolicher — Nijenhuis bracket 33
Nearly complex structure 35
Nijenhuis tensor 35
Nijenhuis — Richardson bracket 28
Nonlinear frame bundle 81
Operational tangent vector 14
Orthonormal frame bundle 53
Orthonormal frame field 52
Parallel transport 39
Principal (fiber) bundle 44
Principal bundle atlas 44
Principal bundle of embeddings 24
Principal connection 60
Principal fiber bundle homomorphism 47
Principal right action 45
Projection 36
Proper 20
Pullback 38
Pullback of fiber bundles 37
Pullback operator 34
Real analytic 12
Real analytic curves 12
Real analytic mappings 12
Recognizing induced connections 67
Reduction of the structure group 47
Regular 23
Restricted holonomy group 65 76
Riemannian metric 52
Right logarithmic derivative 58
Sections of associated bundles 53
Self dual 99
Smooth 6 7
Smooth curves in 18
Smooth mappings 7
Smooth partitions of unity 14
Smoothness of composition 19
Space of connections 84
Standard fiber 36
Stiefel manifold 46
Tangent and vertical bundles of principal and associated bundles 58
Tangent bundles of Grassmann manifolds 56
Tangent bundles of homogeneous spaces 54
Tangent group of a Lie group 57
Tangent vectors as derivations 16
Theorem of Hartogs 10
Topologically real analytic 12
Total space 36
Transformation law for the Christoffel forms 39
Transition functions 36
Universal vector bundle 56
Vector valued differential forms 27
Vertical bundle 37
Vertical projection 37
Vertical space 32
Yang — Mills functional 100
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