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Transcript
Differential geometry of surfaces in Euclidean space
In this short note I would like to illustrate how the general concepts used in Riemannian
geometry arise naturally in the context of curved surfaces in ordinary (Euclidean) space. Let
us assume the n-dimensional space Rn whose coordinates will be labelled with capital Roman
indices, xA . Consider now an m-dimensional (m ≤ n) surface Σ embedded in Rn . It can
be parameterized by a set of m “curvilinear” coordinates, denoted using Greek indices, y µ ;
the surface is defined by giving the Euclidean coordinates xA as a function of the curvilinear
ones, xA = xA (~y ). In the special case m = n, this can be thought of as a formalism to deal
with curvilinear coordinates in Rn . Let us further introduce a basis in the space tangent to
Σ, f~µ . (This basis and the whole tangent space depends on the point ~y , but I will for the
sake of simplicity of notation omit this argument where possible.)
In order to compare vectors tangent to Σ at different points, one needs to define the notion
~ (~y ) ≡ V µ (~y )f~µ (~y ) at a point ~y . Shifting it
of a parallel transport. Take a tangent vector V
paralelly to another, infinitesimally distant point, ~y ′ ≡ ~y + ∆~y , is an intuitively simple task
in Rn : we just keep its coordinates in a given basis defined globally over Rn unchanged.
However, there is a problem, namely that the vector so shifted is not tangent to Σ at ~y ′
anymore. The easiest resolution is project it back to the tangent space of Σ.
In order to transform the above words into formulas, let us further introduce the dual basis
f~µ , defined by f~µ · f~ν = δνµ , where the dot represents the usual scalar product in Rn .1
The dual basis does not coincide with the original basis, unless this is orthonormal. The
projection to the tangent space of Σ is now accomplished using the projection operator
~ from point ~y to ~y ′ then reads
P(~y ) ≡ f~µ (~y )f~µ (~y ). The parallel transport of the vector V
~ (~y ) = f~µ (~y ′)f~µ (~y ′ ) · f~ν (~y )V ν (~y ).
V~⇒~y′ ≡ P(~y ′ ) · V
Now that we can compare tangent vectors at different points of the surface Σ, it also makes
sense to calculate a derivative of a tangent vector along a direction specified by another
~ ; we then define the covariant derivative as
vector. Take the latter to be U
~ ≡ lim
∇U~ V
~)−V
~
V~ (~y + εU
~
⇒~
y +εU
ε
ε→0
.
Let us do the computation step by step,
h
i
~ )V µ (~y + εU)
~ − f~µ (~y + εU)
~ f~µ (~y + εU)
~ · f~ν (~y )V ν (~y ) =
~ = lim 1 f~µ (~y + εU
∇U~ V
ε→0 ε
i
1h
~ ) − f~µ (~y + εU)
~ · f~ν (~y ) V ν (~y ) =
= f~µ (~y ) lim V µ (~y + εU
ε→0 ε |
{z
} |
{z
}
V
µ +εU ν ∂ V µ
ν
(1)
(2)
δνµ +εU λ ∂λ f~µ ·f~ν
= f~µ U ν ∂ν V µ − U λ ∂λ f~µ · f~ν .
Finally, we rewrite the result as
~ = U λ ∂λ V µ + Γµ V ν f~µ ,
∇U~ V
λν
1
Γµλν ≡ −∂λ f~µ · f~ν = f~µ · ∂λ f~ν .
(3)
In order that the dual basis be unambiguously defined for m < n, we have to further demand that it
spans the same m-dimensional vector subspace of Rn as the basis f~µ .
The last relation defines the Christoffel symbol Γµλν , which captures the spatial variations of
the basis: it expresses the µ-th component of the change of f~ν when transported along the
λ-th coordinate.
Coordinate basis
So far, the basis f~µ could be chosen completely arbitrarily in a way unrelated to the coordinate system ~y . From now on we will specialize to the so-called coordinate basis. To that
end, let us first calculate the squared distance of two infinitesimally separated points on the
surface Σ,
∂xA ∂xB µ ν
ds2 ≡ δAB dxA dxB = δAB µ
dy dy ≡ gµν dy µ dy ν ,
(4)
∂y ∂y ν
which defines the induced metric gµν on Σ. The coordinate basis, denoted by ξ~µ in order to
distinguish it from the general basis f~µ , is now defined as
∂xA
~eA ,
ξ~µ ≡
∂y µ
(5)
where ~eA is the canonical orthonormal basis of Rn . The vector ξ~µ is a tangent vector to the
y µ coordinate line and its length is determined by shifting the y µ coordinate by unity. The
coordinate basis is obviously related to the induced metric by gµν = ξ~µ · ξ~ν . The inverse of
the metric, g µν , can be used to lift indices. In particular, the vectors ξ~µ = g µν ξ~ν form a
basis dual to ξ~µ in accordance to the notation introduced above. One further property of the
coordinate basis that we will use is ∂µ ξ~ν = ∂ν ξ~µ as follows immediately from the definition (5).
Consequently, the associated Christoffel symbol is symmetric under the exchange of its lower
indices. This encodes the intuitively clear fact that if we shift two coordinates, y µ and y ν , by
unity, the result does not depend on the order of these two operations. Using this symmetry
and the definition of the metric (4), it is also easy to verify the expression for the Christoffel
symbol in the coordinate basis, well known from Riemannian geometry,
1
Γµλν = ξ~µ · ∂λ ξ~ν = (∂λ gµν + ∂ν gλµ − ∂µ gλν ).
2
(6)
Riemann tensor
The Christoffel symbol is expressed in terms of derivatives of the basis. However, their
appearance may be a mere consequence of an inconvenient choice of the coordinates. The
intrinsic curvature of the surface, independent of such choice, is captured by the Riemann
(curvature) tensor,
Rκλµν = ∂µ Γκνλ − ∂ν Γκµλ + Γκµσ Γσνλ − Γκνσ Γσµλ .
(7)
Note that this expression is valid in any basis so one has to pay attention to the order of
the lower indices in the Christoffel symbol, which is in general not symmetric. Using the
definition of the Christoffel symbol in Eq. (3), we find
Rκλµν = ∂µ f~κ · ∂ν f~λ − ∂ν f~κ · ∂µ f~λ + f~κ · ∂µ f~σ f~σ · ∂ν f~λ − f~κ · ∂ν f~σ f~σ · ∂µ f~λ =
= ∂µ f~κ · ∂ν f~λ − ∂ν f~κ · ∂µ f~λ − ∂µ f~κ · f~σ f~σ · ∂ν f~λ + ∂ν f~κ · f~σ f~σ · ∂µ f~λ .
(8)
Finally, we introduce the projector to the space normal to the surface Σ, P⊥ ≡ 11 − P, to
obtain a simple general expression for the Riemann tensor,
Rκλµν = ∂µ f~κ · P⊥ · ∂ν f~λ − ∂ν f~κ · P⊥ · ∂µ f~λ .
(9)
Note that for m = n the transverse projector P⊥ is zero and the whole Riemann tensor
vanishes as it must for the flat space Rn . Let us now specialize to the coordinate basis.
Here we can lower the first index of the Riemann tensor with the help of the metric gµν
and derive a useful expression for the tensor Rκλµν . We use the fact that in the coordinate
basis the projector P = ξ~µ ξ~µ = g µν ξ~µ ξ~ν as well as P⊥ is a symmetric matrix and hence
ξ~µ · P = P · ξ~µ = ξ~µ and ξ~µ · P⊥ = P⊥ · ξ~µ = 0. Using these properties it is straightforward
to show that gκσ ∂µ ξ~σ · P⊥ = ∂µ ξ~κ · P⊥ so that
Rκλµν = ∂µ ξ~κ · P⊥ · ∂ν ξ~λ − ∂ν ξ~κ · P⊥ · ∂µ ξ~λ.
(10)
Writing P⊥ as P⊥ · P⊥ and using once more the orthogonality of P⊥ and ξ~µ , we arrive at
another expression for the Riemann tensor,
Rκλµν = ξ~κ · ∂µ P · ∂ν P · ξ~λ − ξ~κ · ∂ν P · ∂µ P · ξ~λ .
(11)
Each of the formulas (10) and (11) has its virtues. While the former makes manifest the
antisymmetry of the Riemann tensor in the pairs of indices (κλ) and (µν) as well as the
symmetry with respect to exchange of these pairs, the latter disposes of explicit derivatives
of the basis vectors and thus makes the geometric nature of the Riemann tensor manifest.
Contracting the first and third index in Rκλµν and using once more the symmetry of ∂µ ξ~ν , we
arrive at the corresponding formula for the Ricci tensor and in turn for the scalar curvature,
Rµν = ∂ λ ξ~λ · P⊥ · ∂µ ξ~ν − ∂µ ξ~λ · P⊥ · ∂ν ξ~λ ,
R = ∂ µ ξ~µ · P⊥ · ∂ ν ξ~ν − ∂ µ ξ~ν · P⊥ · ∂µ ξ~ν .
(12)
Surfaces of codimension one
The above derived expressions take a particularly simple form in the special case of surfaces
of codimension one (including the important case of two-dimensional surfaces in R3 ). The
projector P⊥ can then be written in terms of a unit normal vector ~n as P⊥ = ~n~n. Using the
fact that it is orthogonal to all basis vectors ξ~µ , we find the simplified form of the Riemann
tensor,
Rκλµν = (ξ~κ · ∂µ~n)(ξ~λ · ∂ν ~n) − (ξ~κ · ∂ν ~n)(ξ~λ · ∂µ~n).
(13)
All information about the intrinsic curvature of the surface is now encoded in the derivatives
of the normal vector. Using the easily derived identity ξ~µ · ∂ν ~n = ξ~ν · ∂µ~n and the normalization of ~n written as ~n · ∂µ~n = 0, we deduce after a short manipulation the corresponding
expressions for the Ricci tensor and the scalar curvature,
Rµν = (~n · ∂ λ ξ~λ)(~n · ∂µ ξ~ν ) − ∂µ~n · ∂ν ~n,
R = (~n · ∂ µ ξ~µ )2 − ∂µ~n · ∂ µ~n.
(14)