By Maksimov V. I.

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**Extra resources for A Boundary Control Problem for a Nonlinear Parabolic Equation**

**Example text**

In this paper we are mainly focused on the regular curves. See paper [6] for the version of the chain rule which is valid for any ample curve and for basic invariants of unparameterized ample curves. 6 Structural Equations Assume that v and w are two smooth curves in Gn (Σ) such that v(t) ∩ w(t) = 0, ∀t. 4. For any t and any e ∈ v(t) there exists a unique fe ∈ w(t) with the following property: ∃ a smooth curve eτ ∈ v(τ ), et = e, such that d vw : e → ft is linear and for any dτ eτ τ =t = fe . Moreover, the mapping Φt e0 ∈ v(0) there exists a unique smooth curve e(t) ∈ v(t) such that e(0) = e0 and ∀t.

The assumption ker Aτ ∩ker Qτ = 0 implies the smoothness of the mapping (A, Q) → Λ(A, Q) for (A, Q) close enough to (Aτ , Qτ ). 4, this assumption implies that the mapping lef tτ : (ζ, v) → ζAτ + Qτ (v, ·) is surjective. Hence the kernel of the mapping (ζ, v) → ζA + Q(v, ·) (20) smoothly depends on (A, Q) for (A, Q) close to (Aτ , Qτ ). On the other hand, Λ(A, Q) is the image of the mapping (ζ, v) → (ζ, Av) restricted to the kernel of map (20). Now we have to disclose a secret which the attentive reader already knows and is perhaps indignant with our lightness: Qτ is not a well-deﬁned bilinear form on Twτ W, it essentially depends on the choice of local coordinates in M .

N ∈ [E]z be such that ς1 (z), . . , ςn (z) form a basis of Tz Ez . Then ς = b1 ς1 + · · · + bn ςn , where bi are germs of smooth functions vanishing a at z. Commutativity of [E]z implies: 0 = [ςi , ς] = (ςi b1 )ς1 + · · · + (ςi bn )ςn . , bi |Ez = 0, i = 1, . . , n. 5 implies that ς ∈ [E]z is uniquely reconstructed from ς(z). This property permits to deﬁne the vertical derivative of any vertical vector ﬁeld ν on M . Namely, ∀v ∈ Tz Ez we set a Dv ν = [ς, ν](z), where ς ∈ [E]z , ς(z) = v.