calculate the Killing tensors of a specified rank for a given metric or connection

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Tensor[KillingTensors] - calculate the Killing tensors of a specified rank for a given metric or connection

Calling Sequences

KillingTensors(g, p options)

KillingTensors(C, p options)

Parameters

g - a covariant metric tensor on a manifold M

p - a positive integer

C - an affine connection on a manifold M

options - any of the following keywords arguments: ansatz, unknowns, auxiliaryequations, coefficientvariables, parameters, output

Description

•	Let denote covariant differentiation with respect to the given metricor connection A covariant symmetric tensor field of rank is called a Killing tensor if

•

The program KillingTensors generates the defining 1st order partial differential equations for a Killing tensor of rank p and uses pdsolve to find the solution to these equations. An empty list is returned if there are no Killing tensors. If pdsolve is unable to solve these equations, NULL is returned.

•	The keyword argument coefficientvariables allows the user to specify the coefficient functions in the Killing tensor as functions of the variables .

•

The exact form of the Killing tensor can be specified with the keyword argument ansatz . For example, if the coordinates on the underlying manifold are and are defined symmetric tensors, then one may solve for Killing tensors of the form . In this situation the unknown functions must be explicitly specified with the keyword argument unknowns, for example, unknowns

•	When using the keyword argument ansatz, additional algebraic or differential conditions may be imposed upon the unknowns using the keyword argument auxiliaryequations Here is a list of the auxiliary equations to be added to the Killing tensor equations.

•

If the metric or connection C depends upon a number of unspecified parameters (either constants or functions), then the keyword argument parameterswhere is the list of parameters, will invoke case-splitting with respect to these parameters. Special values of the parameters, where either the number or the explicit form of the Killing tensors changes, are calculated.

•

With keyword argument output = the defining partial differential equations for the Killing tensors are returned. The option output = returns the general Killing tensor in terms of a number of arbitray constants , ... . The option output = returns a list of tensors which form a basis for the solution space. The default value of this keyword argument is output =

•	This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form KillingTensors(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-KillingTensors(...).

Examples

Example 1.

Find the Killing tensors of the metric g to order 4 (Note: The fourth order Killing tensor takes a long time to compute). This metric appears in Darboux, Theorie Generale des Surfaces III page 81.

(2.1)

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(2.2)

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(2.3)

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(2.4)

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(2.5)

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[_DG([["tensor", M, [["cov_bas", "cov_bas", "cov_bas", "cov_bas"], []]], [[[1, 1, 1, 1], 1/y^2], [[1, 1, 2, 2], (1/3)/(x*y)], [[1, 2, 1, 2], (1/3)/(x*y)], [[1, 2, 2, 1], (1/3)/(x*y)], [[2, 1, 1, 2], (1/3)/(x*y)], [[2, 1, 2, 1], (1/3)/(x*y)], [[2, 2, 1, 1], (1/3)/(x*y)], [[2, 2, 2, 2], 1/x^2]]])]

(2.6)

Example 2.

We use the keyword argument coefficientvariables to find the rank 3 Killing tensors for the metric g2 which are functions of alone.

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(2.7)

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(2.8)

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(2.9)

Example 3.

We use the keyword arguments ansatz and unknowns to find the rank 2 Killing tensors for the metric g3 which are independent of .

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(2.10)

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(2.11)

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(2.12)

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(2.13)

Example 4.

We use the keyword arguments ansatz, unknowns and auxiliaryequations find the rank 2 Killing tensors for the metric g4 which are invariant under rotations in the plane.

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(2.14)

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(2.15)

We use the commands GenerateSymmmetricTensor and DGzip to construct the general rank 2 symmetric tensor on

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(2.16)

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(2.17)

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_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], A1(x, y, z)], [[1, 2], (1/2)*A2(x, y, z)], [[1, 3], (1/2)*A3(x, y, z)], [[2, 1], (1/2)*A2(x, y, z)], [[2, 2], A4(x, y, z)], [[2, 3], (1/2)*A5(x, y, z)], [[3, 1], (1/2)*A3(x, y, z)], [[3, 2], (1/2)*A5(x, y, z)], [[3, 3], A6(x, y, z)]]])

(2.18)

The vector field is the infinitesimal generator for rotations in the plane.

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(2.19)

We use the commands LieDerivative and DGinfo to find the conditions under which is rotationally invariant.

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(2.20)

We find that that there are 4 rotationally invariant, rank 2 Killing tensors for the metric g4.

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(2.21)

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(2.22)

Example 5.

We wish to determine the rank 2 Killing tensors for the metric for varying values of excluding the case

Because the parameter does not appear as a rational function in it is helpful to re-write the metric as , where satisfies the differential equation

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(2.23)

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(2.24)

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With the keyword argument parameters, the command KillingTensors returns a sequence of lists of Killing tensors and, as the last element of the sequence, the possible exceptional parameter values. For this example, the exceptional values of are:

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$[{m = 1, a(z) = z*_C1}, {m = _C1, a(z) = z^_C1*_C2}]$

(2.25)

We see there are 2 cases.

Case 1.

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(2.26)

Case 2. (the generic case)

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(2.27)

Example 6.

With the keyword argument output = "pde", the defining partial differential equations for the Killing tensor are returned.

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(2.28)

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(2.29)

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(2.30)

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(2.31)

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(2.32)

Example 7.

We compute the Killing tensors for a connection. We use the keyword argument output = "general" to obtain the result as a single tensor depending on constants __ .

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(2.33)

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(2.34)

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(2.35)

Example 8.

The following metric g8 appears in the PhD thesis of R. P. Delong. We are able to explicitly compute all Killing tensors to order 4. The explicit lists are very long and so we simply display the number of Killing tensors at each order.

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