By Maksimov V. I.
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Extra resources for A Boundary Control Problem for a Nonlinear Parabolic Equation
In this paper we are mainly focused on the regular curves. See paper  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.