A naive approach to Tensors on Manifolds Cho, Yong-Hwa Department of Mathematical Sciences, KAIST 1 / 35 Manifolds? 2 / 35 Manifolds? 2 / 35 Manifolds? 2 / 35 = Topological Nature 3 / 35 Topological Nature = 3 / 35 \Flat Space" Rescaling the plane Studying Manifolds: How? 4 / 35 \Flat Space" Rescaling the plane Studying Manifolds: How? 4 / 35 \Flat Space" Rescaling the plane Studying Manifolds: How? 4 / 35 \Flat Space" Studying Manifolds: How? Rescaling the plane 4 / 35 Studying Manifolds: How? \Flat Space" Rescaling the plane 4 / 35 = 126 \Oriented Line density" Vectors and Covectors (2D) · ? = Scalar 5 / 35 Scalar = 12 \Oriented Line density" Vectors and Covectors (2D) · ? = = 6 5 / 35 Scalar = 6 \Oriented Line density" Vectors and Covectors (2D) · ? = = 12 5 / 35 Scalar = 12 Vectors and Covectors (2D) · ? = = 6 \Oriented Line density" 5 / 35 Codimension 1 oriented plane density · = = 3 Dimension 1 oriented plane capacity Vectors and Covectors (3D) 6 / 35 Codimension 1 oriented plane density Dimension 1 oriented plane capacity Vectors and Covectors (3D) · = = 3 6 / 35 Vectors and Covectors (3D) Codimension 1 oriented plane density · = = 3 Dimension 1 oriented plane capacity 6 / 35 Forms 2-form 3-form ! & "= " & = 2-vector 3-vector Multivectors Multivectors and Forms 1-form 1-vector 7 / 35 Forms 3-form " & = 3-vector Multivectors Multivectors and Forms 1-form 2-form ! & "= 1-vector 2-vector 7 / 35 Forms Multivectors Multivectors and Forms 1-form 2-form 3-form ! & "= " & = 1-vector 2-vector 3-vector 7 / 35 Multivectors and Forms Forms 1-form 2-form 3-form ! & "= " & = 1-vector 2-vector 3-vector Multivectors 7 / 35 contravariant parts(vectors) covariant parts(forms) A tensor T with n covariant and m contravariant parts b1b2···bm Ta1a2···an !a !ab !abc v a v ab v abc Abstract Index Notation 8 / 35 contravariant parts(vectors) covariant parts(forms) !a !ab !abc v a v ab v abc Abstract Index Notation A tensor T with n covariant and m contravariant parts b1b2···bm Ta1a2···an 8 / 35 covariant parts(forms) !a !ab !abc v a v ab v abc Abstract Index Notation A tensor T with n covariant and m contravariant parts b1b2···bm contravariant parts(vectors) Ta1a2···an 8 / 35 contravariant parts(vectors) !a !ab !abc v a v ab v abc Abstract Index Notation A tensor T with n covariant and m contravariant parts b1b2···bm Ta1a2···an covariant parts(forms) 8 / 35 Abstract Index Notation A tensor T with n covariant and m contravariant parts b1b2···bm contravariant parts(vectors) Ta1a2···an covariant parts(forms) !a !ab !abc v a v ab v abc 8 / 35 Tensor product - different indicies, indicies determine the order: (~v ⊗ ~w =) v aw b = w bv a 6= v bw a (= ~w ⊗~v) Contraction - same index: a ~v · ! = v !a Contractions between tensors - indicies determine the contraction: c a c a c c (~v · T)b d = v Tab d 6= v Tba d = (~v · T)b d Index Notation, Tensor Products and Contractions 9 / 35 Contraction - same index: a ~v · ! = v !a Contractions between tensors - indicies determine the contraction: c a c a c c (~v · T)b d = v Tab d 6= v Tba d = (~v · T)b d Index Notation, Tensor Products and Contractions Tensor product - different indicies, indicies determine the order: (~v ⊗ ~w =) v aw b = w bv a 6= v bw a (= ~w ⊗~v) 9 / 35 Contractions between tensors - indicies determine the contraction: c a c a c c (~v · T)b d = v Tab d 6= v Tba d = (~v · T)b d Index Notation, Tensor Products and Contractions Tensor product - different indicies, indicies determine the order: (~v ⊗ ~w =) v aw b = w bv a 6= v bw a (= ~w ⊗~v) Contraction - same index: a ~v · ! = v !a 9 / 35 Index Notation, Tensor Products and Contractions Tensor product - different indicies, indicies determine the order: (~v ⊗ ~w =) v aw b = w bv a 6= v bw a (= ~w ⊗~v) Contraction - same index: a ~v · ! = v !a Contractions between tensors - indicies determine the contraction: c a c a c c (~v · T)b d = v Tab d 6= v Tba d = (~v · T)b d 9 / 35 = = a = v !ab = ~v · ! · = = 6 a ~v · ! = v !a · = = 9 ~ = 1 ab ~v · ! 2! v !ab · ~v · ! Tensor Algebra: Contractions of co and contra parts 10 / 35 = = a = v !ab = ~v · ! · = = 9 ~ = 1 ab ~v · ! 2! v !ab · ~v · ! Tensor Algebra: Contractions of co and contra parts · = = 6 a ~v · ! = v !a 10 / 35 = = a = v !ab = ~v · ! · ~v · ! Tensor Algebra: Contractions of co and contra parts · = = 6 a ~v · ! = v !a · = = 9 ~ = 1 ab ~v · ! 2! v !ab 10 / 35 = = a = v !ab = ~v · ! Tensor Algebra: Contractions of co and contra parts · = = 6 a ~v · ! = v !a · = = 9 ~ = 1 ab ~v · ! 2! v !ab · ~v · ! 10 / 35 = = ~v · ! Tensor Algebra: Contractions of co and contra parts · = = 6 a ~v · ! = v !a · = = 9 ~ = 1 ab ~v · ! 2! v !ab · = a ~v · ! = v !ab 10 / 35 Tensor Algebra: Contractions of co and contra parts · = = 6 a ~v · ! = v !a · = = 9 ~ = 1 ab ~v · ! 2! v !ab · = = a ~v · ! = v !ab = ~v · ! 10 / 35 ^ = = a b a?b b a (v ^ w)ab v ^ w = v w − v w = ^ = ? = !a ^ σb = !aσb − !bσa = (! ^ σ)ab Tensor Algebra: Exterior Product 11 / 35 = a b a?b b a (v ^ w)ab v ^ w = v w − v w = ? = !a ^ σb = !aσb − !bσa = (! ^ σ)ab Tensor Algebra: Exterior Product ^ = ^ = 11 / 35 a b a?b b a (v ^ w)ab v ^ w = v w − v w = ? = !a ^ σb = !aσb − !bσa = (! ^ σ)ab Tensor Algebra: Exterior Product ^ = = ^ = 11 / 35 a b a?b b a (v ^ w)ab v ^ w = v w − v w = ? !a ^ σb = !aσb − !bσa = (! ^ σ)ab Tensor Algebra: Exterior Product ^ = = ^ = = 11 / 35 ? ? Tensor Algebra: Exterior Product ^ = = ab v a ^ w b = v aw b − v bw a = (v ^ w) ^ = = !a ^ σb = !aσb − !bσa = (! ^ σ)ab 11 / 35 !a ^ σbc = !aσbc + !bσca + !cσab = (! ^ σ)abc Tensor Algebra: Exterior Product k-form ^ l-form = (k + l)-form ^ = = 12 / 35 Tensor Algebra: Exterior Product k-form ^ l-form = (k + l)-form ^ = = !a ^ σbc = !aσbc + !bσca + !cσab = (! ^ σ)abc 12 / 35 3 2 1 Tensor Fields: Scalar and Covector Fields 13 / 35 3 2 1 Tensor Fields: Scalar and Covector Fields 13 / 35 Tensor Fields: Scalar and Covector Fields 3 2 1 13 / 35 Space Compression r Incompatible r Example: Gradient field rφ Space Compression r~ r~ 14 / 35 Space Compression r r Example: Gradient field rφ Space Compression r~ Incompatible r~ 14 / 35 Incompatible Space Compression r~ r~ Example: Gradient field rφ Space Compression r r 14 / 35 r^ r^ = r ^ ! r^ = r ^ ! r ^ r^ = 0 Tensor Calculus: Exterior Differentiation r^ (=d) 15 / 35 = r ^ ! = r ^ ! r ^ r^ = 0 Tensor Calculus: Exterior Differentiation r^ (=d) r^ r^ r^ 15 / 35 r ^ ! r ^ ! r ^ r^ = 0 Tensor Calculus: Exterior Differentiation r^ (=d) r^ r^ = r^ = 15 / 35 = = r ^ ! r ^ r^ = 0 r^ Tensor Calculus: Exterior Differentiation r^ (=d) r^ r ^ ! 15 / 35 r^ = r ^ ! = r ^ ! r ^ r^ = 0 r^ Tensor Calculus: Exterior Differentiation r^ (=d) r^ 15 / 35 r^ r ^ ! r ^ ! r ^ r^ = 0 Tensor Calculus: Exterior Differentiation r^ (=d) r^ r^ = r^ = 15 / 35 = r ^ ! = r ^ r^ = 0 r^ r^ Tensor Calculus: Exterior Differentiation r^ (=d) r^ r ^ ! 15 / 35 = r ^ ! = r ^ ! r ^ r^ = 0 r^ r^ Tensor Calculus: Exterior Differentiation r^ (=d) r^ 15 / 35 r ^ ! r ^ ! Tensor Calculus: Exterior Differentiation r^ (=d) r^ r^ = r^ = r ^ r^ = 0 15 / 35 Z ! = 2 L Z ! = 3 S Tensor Calculus: Integration Integration of k-form fields over oriented k-dimensional surfaces: 16 / 35 Tensor Calculus: Integration Integration of k-form fields over oriented k-dimensional surfaces: Z ! = 2 L Z ! = 3 S 16 / 35 0+1+1−1+1−1+1 = 2 +1−1+1+0+1−1+1 = 2 Z Z r ^ k ! = k ! Sk+1 @Sk Stokes' Theorem Z r ^ ! = S Z ! = @S 17 / 35 +1+1−1+1−1+1 = 2 +1+0+1−1+1 = 2 Z Z r ^ k ! = k ! Sk+1 @Sk Stokes' Theorem +1 Z −1 r ^ ! = 0 S Z ! = +1−1 @S 17 / 35 +1−1+1−1+1 = 2 +0+1−1+1 = 2 Z Z r ^ k ! = k ! Sk+1 @Sk Stokes' Theorem +1 +1 Z r ^ ! = 0+1 S Z ! = +1−1+1 @S 17 / 35 +1−1+1 = 2 +1−1+1 = 2 Z Z r ^ k ! = k ! Sk+1 @Sk Stokes' Theorem +1 Z r ^ ! = 0+1+1−1 −1 S Z ! = +1−1+1+0 @S 17 / 35 +1 = 2 +1 = 2 Z Z r ^ k ! = k ! Sk+1 @Sk Stokes' Theorem Z r ^ ! = 0+1+1−1+1−1 S Z ! = +1−1+1+0+1−1 −1 +1 @S −1 +1 17 / 35 Z Z r ^ k ! = k ! Sk+1 @Sk Stokes' Theorem Z r ^ ! = 0+1+1−1+1−1+1 = 2 S Z ! = +1−1+1+0+1−1+1 = 2 +1 @S +1 17 / 35 Stokes' Theorem Z r ^ ! = 0+1+1−1+1−1+1 = 2 S Z ! = +1−1+1+0+1−1+1 = 2 @S Z Z r ^ k ! = k ! Sk+1 @Sk 17 / 35 the volume form n a1···an and the volume element n~ -1 1 · n~ -1= ( -1)a1···an = 1 n n! a1···an provide the unit density, volume and the space orientation. The volume of the manifold M: Z V (M) = n M Volume Form and Volume Element On the n dimensional orientable manifold, 18 / 35 The volume of the manifold M: Z V (M) = n M Volume Form and Volume Element On the n dimensional orientable manifold, the volume form n a1···an and the volume element n~ -1 1 · n~ -1= ( -1)a1···an = 1 n n! a1···an provide the unit density, volume and the space orientation.