Gravitating global defects: The gravitational field and compactification

We give a prescription to add the gravitational field of a global topological defect to a solution of Einstein’s equations in an arbitrary number of dimensions. We only demand that the original solution have an $O\left(n\right)\mathrm{}$ invariance with $n>~3.$ We will see that the general effect of a global defect is to introduce a deficit solid angle. We also show how the same kind of scalar field configurations can be used for spontaneous compactification of n extra dimensions on an n-sphere.