Riemann curvature tensor has four symmetries. Therefore, the total number of algebraically independent components of the curvature tensor in N dimensions is . Notice that e = ε + !. Metrics details. The condition of invariance reduces the number of the independent tensor components, since it signifies relationships between the tensor components. The number of independent components of tensors in symmetrical systems. Minimize the number of tensor components according to its symmetries (and relabel, redefine or count the number of independent tensor components) The nice development described below is work in collaboration with Pascal Szriftgiser from Laboratoire PhLAM, Université Lille 1, France, used in the Mapleprimes post Magnetic traps in cold-atom physics . Also, these components can be arranged in sets of three that satisfy the three-way skew symmetry, so the number of independent components of this form is reduced by a factor of 2/3. For a tensor of higher rank ijk lA if ijk jik l lA A is said to be symmetric w.r.t the indices i,j only . The expression "independent components of a tensor" is misleading. PDF File (170 KB) Article info and citation; First page; Article information. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. This logic can be extended to see that in an N-dimensional space, a tensor of rank R can have N^R components. There is one final symmetry condition for the Riemann tensor, and it is the trickiest to handle. A symmetric tensor of rank 2 in N-dimensional space has ( 1) 2 N N independent component Eg : moment of inertia about XY axis is equal to YX axis . In other words, the tensor representation of the rotation group is actually reducible – it breaks up into a six component and a three component WDGO. The 21 components are not all independent, though - it turns out that there is a constraint equation from the Bianchi identies. 4 . Thus, like stress, strain is by definition a symmetric tensor and has only 6 independent components. And the total number of independent components in four-dimensional spacetime is therefore 21-1 = 20 independant components. Source Bull. How is this symmetricity going to affect the number of components? The number of suffixes is the rank of the Cartesian tensor, a ... which is an antisymmetric tensor has three independent components which are the vector components of the vector product U → × V →, and the third term is a symmetric traceless tensor, which has five independent components. We are left with (3.85) independent components of the Riemann tensor. Altogether, then, there are 1 + 3 + 5 = 9 components, as required. So we can say that [math]A^{ij}=-A^{ji}[/math]. A Riemannian space is a manifold characterized by the existing of a symmetric rank-2 tensor called the metric tensor. For a totally antisymmetric vector with rank rand aantisymmetric components in a n-folds, we have already shown that the number of independent components is given by: nr a n! (n a)!a! once time that 0123 is given, the tensor is xed in an unique way. Tensors of rank 2 or higher that arise in applications usually have symmetries under exchange of their slots. So the number of degrees of freedom drops from 21 down to 20, because of this constraint. 1.11.2 Real Symmetric Tensors Suppose now that A is a real symmetric tensor (real meaning that its components are real). This is precisely the number of independent components of the Riemann tensor, which we will de ne and study later on, and which is a measure of the curvature of spacetime. Soc., Volume 49, Number 6 (1943), 470-472. Therefore, the number of independent terms in the curvature tensor becomes n2(n 21) 2=4 n2(n 1)(n 2)=6 = n (n 1)=12. 9.5.1 Symmetry by Definition Some properties are defined such that the corresponding tensors exhibit an inner symmetry. number of independent components from 256 to 20. References (1) F.G. Fumi:Nuovo Cimento,9, 739 (1952): R. Fieschi and F. G. Fumi:Nuovo Ciinento,10, 865 (1953). Symmetric Tensor : T λµ= T µλorT ( ), T νλµ= T νµλor T ν(λµ) Antisymmetric : T λµ= −T µλor T [ ], T νλµ= −T νµλor T ν[λµ] Number of independent components : Symmetric: n(n + 1)/2, Antisymmetric: n(n −1)/2 9. My prof just acted like I should be able to do this in my sleep, but I am struggling. p{ the number comes from the symmetry of the partial derivatives. In theories and experiments involving physical systems with high symmetry, one frequently encounters the question of how many independent terms are required by symmetry to specify a tensor of a given rank for each symmetry group. 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