A Comprehensive Introduction to Differential Geometry, Volume 1Publish or Perish, Incorporated, 1975 - Geometry, Differential |
From inside the book
Results 1-3 of 29
Page 4-12
... covariant vector fields and superscripts a for contravariant vector fields . After this was firmly established , the indices on the x's were shifted upstairs again to make the summation convention work out . ) Covariant and ...
... covariant vector fields and superscripts a for contravariant vector fields . After this was firmly established , the indices on the x's were shifted upstairs again to make the summation convention work out . ) Covariant and ...
Page 4-14
... covariant tensors of order k , and a section is called a covariant tensor field of order If ( x , U ) is a coordinate system , so that dx1 ( p ) , ... , dx " ( p ) is a basis for ( M ) * , then the k - fold tensor products dx11 ( p ) ...
... covariant tensors of order k , and a section is called a covariant tensor field of order If ( x , U ) is a coordinate system , so that dx1 ( p ) , ... , dx " ( p ) is a basis for ( M ) * , then the k - fold tensor products dx11 ( p ) ...
Page 4-18
... covariant vector field vector fields , a a c map f : MN gives a map w . Just as with covariant f * taking covariant tensor fields A of order k on N to covariant tensor fields f * A of order k on M : f.X f * A ( p ) ( X ̧ ̧ , 4-18.
... covariant vector field vector fields , a a c map f : MN gives a map w . Just as with covariant f * taking covariant tensor fields A of order k on N to covariant tensor fields f * A of order k on M : f.X f * A ( p ) ( X ̧ ̧ , 4-18.
Other editions - View all
Common terms and phrases
ax¹ called choose clearly compact support components compute connected consider continuous coordinate system coordinate system x,U COROLLARY countable covariant critical point define definition denote diffeomorphism differential equations dt dt dx¹ element exp tY fibre finite function f geodesic homeomorphic identity inner product integral curve integral manifold isomorphism k-form left invariant Lemma Let f Lie algebra Lie group linear M₁ M₂ manifold-with-boundary map f matrix Möbius strip n-dimensional n-manifold neighborhood obtain one-one open set order isomorphic orientation preserving Problem Proof Proposition prove Riemannian metric satisfies sequence Show structure subgroup submanifold subset subspace Suppose tangent bundle tangent vector tensor field Theorem topology U₁ unique V₁ vector bundle vector field vector space w₂ θα Ρε Σα χε ду дх