Unique Derivative Operator: Reparametrization. Metric Part 2.

Moving on from first part.

Suppose ∇ is a derivative operator, and g_ab is a metric, on the manifold M. Then ∇ is compatible with g_ab iff ∇_a g_bc = 0.

Suppose γ is an arbitrary smooth curve with tangent field ξ^a and λ^a is an arbitrary smooth field on γ satisfying ξⁿ∇_nλ^a = 0. Then

ξⁿ∇_n(g_abλ^aλ^b) = g_abλ^aξⁿ ∇_nλ^b + g_abλ^bξⁿ ∇_nλ^a + λ^aλ^bξⁿ ∇_ng_ab

= λ^aλ^bξⁿ∇_ng_ab

Suppose first that ∇_ng_ab = 0. Then it follows immediately that ξⁿ∇_ng_abλ^aλ^b = 0. So ∇ is compatible with g_ab. Suppose next that ∇ is compatible with g_ab. Then ∀ choices of γ and λ^a (satisfying ξⁿ∇_nλ^a =0), we have λ^aλ^bξⁿ∇_ng_ab = 0. Since the choice of λ^a (at any particular point) is arbitrary and g_ab is symmetric, it follows that ξⁿ∇_ng_ab = 0. But this must be true for arbitrary ξ_a (at any particular point), and so we have ∇_ng_ab = 0.

Note that the condition of compatibility is also equivalent to ∇_ag^bc = 0. Hence,

0 = g^bn∇_aδ^c_n = g^bn∇_a(g_nrg^rc) = g^bng_nr∇_ag^rc + g^bng^rc∇_ag_nr

= δ^b_r∇_ag^rc + g^bng^rc∇_ag_nr = ∇_ag^bc + g^bng^rc∇_ag_nr.

So if ∇_ag_bc = 0,it follows immediately that ∇_ag^bc = 0. Conversely, if ∇_ag^bc =0, then g^bng^rc∇_ag_nr = 0. And therefore,

0 = g_pbg_scg^bng^rc∇_ag_nr = δⁿ_pδ^r_s∇_ag_nr = ∇_ag_ps

The basic fact about compatible derivative operators is the following.

Suppose g_ab is a metric on the manifold M. Then there is a unique derivative operator on M that is compatible with g_ab.

It turns out that if a manifold admits a metric, then it necessarily satisfies the countable cover condition. And then it guarantees the existence of a derivative operator.) We do prove that if M admits a derivative operator ∇, then it admits exactly one ∇′ that is compatible with g_ab.

Every derivative operator ∇′ on M can be realized as ∇′ = (∇, C^a_bc), where C^a_bc is a smooth, symmetric field on M. Now

∇′_ag_bc = ∇_ag_bc + g_nc Cⁿ_ab + g_bn Cⁿ_ac = ∇_ag_bc + C_cab + C_bac. So ∇′ will be compatible with g_ab (i.e., ∇′_ag_bc = 0) iff

∇_ag_bc = −C_cab − C_bac —– (1)

Thus it suffices for us to prove that there exists a unique smooth, symmetric field C^a_bc on M satisfying equation (1). To do so, we write equation (1) twice more after permuting the indices:

∇_cg_ab = −C_bca − C_acb,

∇_bg_ac = −C_cba − C_abc

If we subtract these two from the first equation, and use the fact that C_abc is symmetric in (b, c), we get

C_abc = 1/2 (∇_ag_bc − ∇_bg_ac − ∇_cg_ab) —– (2)

and, therefore,

C^a_bc = 1/2 g^an (∇_ng_bc − ∇_bg_nc − ∇_cg_nb) —– (3)

This establishes uniqueness. But clearly the field Cabc defined by equation (3) is smooth, symmetric, and satisfies equation (1). So we have existence as well.

In the case of positive definite metrics, there is another way to capture the significance of compatibility of derivative operators with metrics. Suppose the metric g_ab on M is positive definite and γ : [s₁, s₂] → M is a smooth curve on M. We associate with γ a length

|γ| = ∫_s1^s₂ g_abξ^aξ^b ds,

where ξ^a is the tangent field to γ. This assigned length is invariant under reparametrization. For suppose σ : [t₁, t₂] → [s₁, s₂] is a diffeomorphism we shall write s = σ(t) and ξ′^a is the tangent field of γ′ = γ ◦ σ : [t₁, t₂] → M. Then

ξ′^a = ξ^ads/dt

We may as well require that the reparametrization preserve the orientation of the original curve – i.e., require that σ (t₁) = s₁ and σ (t₂) = s₂. In this case, ds/dt > 0 everywhere. (Only small changes are needed if we allow the reparametrization to reverse the orientation of the curve. In that case, ds/dt < 0 everywhere.) It

follows that

|γ’| = ∫_t1^t₂ (g_abξ′^aξ′^b)^1/2 dt = ∫_t1^t₂ (g_abξ^aξ^b)^1/2 ds/dt

= ∫_s1^s₂ (g_abξ^aξ^b)^1/2 ds = |γ|

Let us say that γ : I → M is a curve from p to q if I is of the form [s₁, s₂], p = γ(s₁), and q = γ(s₂). In this (positive definite) case, we take the distance from p to q to be

d(p,q)=g.l.b. |γ|:γ is a smooth curve from p to q.

Further, we say that a curve γ : I → M is minimal if, for all s ∈ I, ∃ an ε > 0 such that, for all s₁, s₂ ∈ I with s₁ ≤ s ≤ s₂, if s₂ − s₁ < ε and if γ′ = γ|[s₁, s₂] (the restriction of γ to [s₁, s₂]), then |γ′| = d(γ(s₁), γ(s₂)) . Intuitively, minimal curves are “locally shortest curves.” Certainly they need not be “shortest curves” outright. (Consider, for example, two points on the “equator” of a two-sphere that are not antipodal to one another. An equatorial curve running from one to the other the “long way” qualifies as a minimal curve.)

One can characterize the unique derivative operator compatible with a positive definite metric g_ab in terms of the latter’s associated minimal curves. But in doing so, one has to pay attention to parametrization.

Let us say that a smooth curve γ : I → M with tangent field ξ^a is parametrized by arc length if ∀ ξ^a, g_abξ^aξ^b = 1. In this case, if I = [s₁, s₂], then

|γ| = ∫_s1^s₂ (g_abξ^aξ^b)^1/2 ds = ∫_s1^s₂ 1.ds = s₂ – s₁

Any non-trivial smooth curve can always be reparametrized by arc length.

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