Extrinsic Curvature as Geodesics Deviation

i.e., Leibniz rule. And of course the operation d δ should be excluded from the wedge’s. We
use a bracket on δ as d (δ) to express this fact explicitly.
Before we continue, let’s introduce some notations
C∑
N µν = g µν − h µν =
(I) ɛ (I) n (I) µ n ν . (22)
I=1
(I) N µν = N µν − (I) ɛ (I) n µ (I) n ν =
C∑
(J) ɛ (J) n µ (J) n ν . (23)
J=1,J≠I
We notice that (I) n ν (I) n ν = (I) ɛ = const along Σ, thus we have
(
d δ (I) n ν
)
(I)
n ν = 0, (24)
which implies
d (I) δ n ν = hδ α ∇ α (I) n ν =
=
( )
hδ α ∇ α (I) n µ
( )
hδ α ∇ α (I) n µ (h ν µ + N ν µ ) =
g ν
µ
( ) (
hδ α ∇ α (I) n µ h ν µ + (I) N ν
µ
)
, (25)
where we have used equation (24) in deriving the last equal sign. The above equation is of
course just
where we have defined
d δ (I) n ν = (I) K δν +
(IJ) B δ ≡
C∑
J=1,J≠I
(J) ɛ (J) n ν (IJ) B δ , (26)
(
(J)
n µ d δ (I) n µ
)
. (27)
Overall, the above procedure is nothing but projecting d δ (I) n ν to the quantity lying on Σ
and the quantity normalizing to Σ.
Now we can use the fact
and equation (26) to rewrite equation (21) as
K δµβ...ν = (1) K (δ)µ ∧ (2) n β ∧ ... ∧ (C) n ν
... ∧ (J) n ν ∧ ... ∧ (J) n µ ∧ ... = 0, (28)
+ (1) n µ ∧ (2) K (δ)β ∧ ... ∧ (C) n ν + (1) n µ ∧ (2) n β ∧ ... ∧ (C) K (δ)ν . (29)
Again, when we do wedge, the δ is not affected. It is apparent that the terms above are
linearly independent, thus K δµβ...ν is a tensor whose coefficients of linearly independent tensors
are (I) K αβ . In another word, there is a one-to-one correspondence between K δµβ...ν and
a set of (I) K αβ 4 . Therefore, NVP extrinsic curvature K δµβ...ν is equivalent to C quantities
(I) K µν , which characterize extrinsic curvature under GEP.
Now we can conclude that GEP and NVP are equivalent for any codimension.
4 The above discuss can be clearer by the notation of [2], Actually this is how I found the way to solve it.
In [2] tangent space and cotangent space are not distinguished (this is OK when metric tensor is defined on
the manifold since metric defines an one-to-one map between tangent and cotangent space.) and vectors are
4