- #1

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## Main Question or Discussion Point

Our 4-D space is ##x^1,x^2,x^3 ,t##.

Our sub-manifold is defined by ##(x^1,x^2,x^3)##

Therefore for this sub-manifold to be maximally symmetric and for which the tangent vector ##\frac{∂}{∂t}(\hat t)## orthogonal to this sub-manifold

The metric becomes,

##ds^2=g(t)dt^2+f(t)(dr^2+r^2d\Omega^2)##

From the known metric for 4-D space and comparing this with above ##g(t)=-1## and for static ##f(t)=1## and for expanding ##f(t)=a(t)^2## ,is it the case ??

And can we say that the hypersurface to any t is orthogonal to the increament direction of t??

Our sub-manifold is defined by ##(x^1,x^2,x^3)##

Therefore for this sub-manifold to be maximally symmetric and for which the tangent vector ##\frac{∂}{∂t}(\hat t)## orthogonal to this sub-manifold

The metric becomes,

##ds^2=g(t)dt^2+f(t)(dr^2+r^2d\Omega^2)##

From the known metric for 4-D space and comparing this with above ##g(t)=-1## and for static ##f(t)=1## and for expanding ##f(t)=a(t)^2## ,is it the case ??

And can we say that the hypersurface to any t is orthogonal to the increament direction of t??