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Unit 12. Curvature ====== Unit 12. Curvature ========================================================================================== ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Curvature operator, curvature tensor, Bianchi identities, Riemann-Christoffel tensor, symmetry properties of the Riemann-Christoffel tensor, sectional curvature, Schur’s theorem, space forms, Ricci tensor, Ricci curvature, scalar curvature, curvature tensor of a hypersurface. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- If D is an affine connection on a manifold M, then we may consider the operator R(X,Y) = [DX,D Y]- D [X,Y]:X(M)----------L X(M), where [DXYXYYX,D ]=D qD -D qD is the usual commutator of operators. The mapping that assigns to the vector fields X,Y the operator R(X,Y) is called the ------------------------------------------------------------------------------------------curvature operator of the connection. The assignment X(M)xX(M)xX(M)----------L X(M) (X,Y,Z)9-----L R(X,Y)(Z) is called the-------------------------------------------------------------------------------- curvature tensor of the connection. To reduce the number of brackets, we shall denote R(X,Y)(Z) simply by R(X,Y;Z). Thus, the letter R is used in two different meanings, later it will denote also a third mapping, but the number of arguments of R makes always clear which meaning is considered. -------------------------------------------------------Proposition. The curvature tensor is linear over the ring of smooth functions in each of its arguments, and it is skew symmetric in the first two arguments. -------------------------Proof. Skew symmetry in the first two arguments is clear, since R(X,Y) = [DX,D Y]- D [X,Y]= -[D Y,D X] + D [Y,X] = -R(Y,X). According to this, it suffices to check linearity of the curvature tensor in the first and third arguments. Linearity in the first argument is proved by the following identities. R(X +X ,Y) = [D ,D ]- D = [D +D ,D ]- D = 1 2 X1 +X 2 Y [X 1 +X 2 ,Y] X 1 X 2 Y [X 1 ,Y]+[X 2 ,Y] = [D ,D ] + [D ,D ]- D - D = R(X ,Y) + R(X ,Y). X1 Y X 2 Y [X 1 ,Y] [X 2 ,Y] 1 2 and R(fX,Y;Z) = ([DfX,D Y]- D [fX,Y])(Z) = fDD X YZ-D Y(fD XZ)-D f[X,Y]-Y(f)X(Z) = = fDDX YZ - fDD Y XZ - Y(f)D XZ -fD [X,Y]Z + Y(f)D X(Z) = = f(DDX YZ- DD Y XZ- D [X,Y]Z) = f R(X,Y;Z). 1 Additivity in the third argument is clear, since R(X,Y) is built up of the additive operators DXY, D and their compositions. To have linearity, we need R(X,Y;fZ) = DDX Y(fZ) - DD Y X(fZ) - D [X,Y](fZ) = = DX(Y(f)Z+fD YZ)-D Y(X(f)Z+fD XZ)-[X,Y](f)Z-fD [X,Y]Z = = XY(f)Z+Y(f)DXYXYYXYXZ+X(f)D Z+fDD Z -YX(f)Z-X(f)D Z-Y(f)D Z-fDD Z- -XY(f)Z+YX(f)Z - fD[X,Y]Z = f(DDX YZ- DD Y XZ- D [X,Y]Z) = f R(X,Y;Z).44444 The proposition is a bit surprising, because the curvature tensor is built up from covariant derivations, which are not linear operators over the ring of smooth functions. We have already introduced tensor fields over a hypersurface. We can introduce tensor fields over a manifold in the same manner. A tensor field T of type (k,l) is an assignment to every point p of a manifold M a tensor T(p) of type (k,l) over the tangent space Tp M. If d 1,...,d n are the basis vector fields defined by a chart over the domain of the chart, and we denote by 1 n dx (p), ..., dx (p) the dual basis of d1(p),...,d n(p), then a tensor field is uniquely determined over the domain of the chart by the components i ...i i i T 1 k(p)=T(p)(dx 1 (p),...,dx k (p);d (p),...,d (p)). j ...j j j 1 l 1 l We say that the tensor field is------------------------------ smooth, if for any chart the functions i ...i T 1 k are smooth. We shall consider only smooth tensor fields. j ...j 1 l Tensors of valency (1,0) are the vector fields, tensors of valency (0,1) are the---------------------------------------------------------------------------------------------------- differential 1-forms. Thus, a differential 1-form assigns to every point of the manifold a linear function on the tangent space at that point. Differential 1-forms form a module over the ring of smooth functions, which we denote by W1(M). Every tensor field defines a multi-F(M)-linear mapping W1(M)x...xW 1(M)xX(M)x...xX(M) ----------L F(M) and conversely, every such multi-F(M)-linear mapping comes from a tensor field. (Check this!) Therefore, tensor fields can be identified with multi- F(M)-linear mappings W1(M)x...xW 1(M)xX(M)x...xX(M) ----------L F(M). Tensors of type (1,k), that is multi-F(M)-linear mappings W1(M)xX(M)x...xX(M) ----------L F(M) can be identified in a natural way with multi-F(M)-linear mappings X(M)x...xX(M) ----------L X(M). 2 By this identification, R:X(M)x...xX(M) ----------L X(M) corresponds to ~ 1 ~ R:W (M)xX(M)x...xX(M) ----------L F(M), defined by R(w;X1 ,...,X k )=w(R(X 1 ,...,X k )). Using these identifications, the curvature tensor is a tensor field of valency (1,3) by the proposition. It is a remarkable consequence, that although the vectors DXYZ(p) and D Z(p) are not determined by the vectors X(p),Y(p),Z(p), to compute the value of R(X,Y;Z) at p it suffices to know X(p),Y(p),Z(p). Beside skew-symmetry in the first two arguments, the curvature tensor has many other symmetry properties. Theorem. (First Bianchi Identity). If R is the curvature tensor of a torsion free connection, then R(X,Y;Z) + R(Y,Z;X) + R(Z,X;Y) = 0 for any three vector fields X,Y,Z. -------------------------Proof. Let us introduce the following notation. If F(X,Y,Z) is a function of the vector fields X,Y,Z, then denote byuJo F(X,Y,Z) or uJo F(X,Y,Z) the sum m-----.XYZ m-----. of the values of F at all cyclic permutations of the variables (X,Y,Z) uJo m-----.F(X,Y,Z) = F(X,Y,Z) + F(Y,Z,X) + F(Z,X,Y). uJo We shall use several times that behind the cyclic summationm-----. we may cyclically rotate X,Y,Z in any expression uJo uJo uJo m-----.F(X,Y,Z) = m-----. F(Y,Z,X) = m-----. F(Z,X,Y). The theorem claims vanishing of uJo uJo uJo m-----.R(X,Y;Z) = m-----. (DDX YZ- DD Y XZ- D [X,Y]Z) =m-----. (DDX YZ- DD X ZY- D [X,Y]Z) uJo uJo uJo =m-----. (DX[Y,Z] - D [X,Y]Z) =m-----. (DZ[X,Y] - D [X,Y]Z) =m-----. [Z,[X,Y]], but the latter expression is 0 according to the Jacobi identity on the Lie bracket of vector fields. (At the third and fifth equality we used the torsion free property of D.) 44444 The presence of an affine connection on a manifold allows us to differentiate not only vector fields, but also tensor fields of any type. Definition. Let (M,D) be a manifold with an affine connection. If --------------------------------------------------1 w e W (M) is a 1-form, X is a vector field, then we define the--------------------------------------------- covariant derivative-------------------------------------------------- DXw of--------------------w with respect to X to be the 1-form (DXXw)(Y) = X(w(Y)) - w(D Y), YeX(M). In general, the---------------------------------------------------------------------------------------------------- covariant derivative DXT -------------------------------------------------------------------------------------of a tensor field T:W1(M)x...xW 1(M)xX(M)x...xX(M) ----------L F(M) of valency (k,l) with respect to a vector field X is a tensor field of the 3 same valency, defined by (DXT)(w 1,...,w k;X 1 ,...,Xl ) = X(T(w1,...,w k;X 1 ,...,Xl )) - k - S T(w ,...,D w ,...,w ;X ,...,X ) - i=1 1 X i k 1 l l - S T(w ,...,w ;X ,...,D X ,...,X ). j=1 1 k 1 X j l For the case of the curvature tensor, this definition gives (DXXXXXR)(Y,Z;W) = D (R(Y,Z;W)-R(D Y,Z;W)-R(Y,D Z;W)-R(Y,Z;D W). -----------------------------------Theorem. (Second Bianchi Identity) The curvature tensor of a torsion free connection satisfies the following identity uJo (D R)(Y,Z;W) = (D R)(Y,Z;W) + (D R)(Z,X;W) + (D R)(X,Y;W) = 0. XYZm-----. XXYZ -------------------------Proof. (DXR)(Y,Z;W) is the value of the operator DXXXX qR(Y,Z)-R(D Y,Z)-R(Y,D Z)-R(Y,Z)qD : X(M) ----------L X(M) on the vector field W, hence we have to prove vanishing of the operator uJo D qR(Y,Z)-R(D Y,Z)-R(Y,D Z)-R(Y,Z)qD . XYZm-----. XXXX First, we have uJoD qR(Y,Z)-R(Y,Z)qD = uJo (DDD -DDD -DD )-(DDD -DDD -DD )= XYZm-----. XX m-----. XYZ X Y Z X Z Y X [Y,Z] Y Z X Z Y X [Y,Z] X uJo(DDD -DDD -DD )-(DDD -DDD -DD )=uJo DD - DD . XYZm-----. X Y Z X Z Y X [Y,Z] X Y Z X Z Y [Y,Z] X m-----.XYZ [Y,Z] X X [Y,Z] On the other hand, uJo-R(D Y,Z)-R(Y,D Z) = uJo R(D Z,Y) - R(D Y,Z) = uJo R(D X,Z) - R(D Y,Z) = m-----.XYZXX m-----.
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