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18 i) A curve is a geodesic if and only if the work is zero. 38) becomes ωφ (V ) = ∇φ˙ ωφ (V ) − ωφ (∇φ˙ V ), which shows that the work wφ measures the non-commutativity between ω and ∇φ˙ . 19 Consider the potential function U ∈ F(M). 40) is called the force vector field. 20. 35), iff φ verifies Newton’s equation ∇φ˙ φ˙ = −∇U. 42) Proof. 44) 1 ˙ φ). 44) becomes ∂U − = 0. ∂xk Multiplying by g kl , summing over k, and adding the last two equations, we find φ¨ l + l is | φ(t) φ˙ i φ˙ s = −gkl which is the Euler–Lagrange equation for L.

24 If U is constant on M, we get the well-known result that the vector tangent to a geodesic has a constant length. The Total Energy Even when there are no Killing vectors on M, we can always find another first integral of motion, called total energy: E(φ) = 1 ˙ ˙ ) + U φ(t) . 51) E is the sum of the kinetic and the potential energy, while the Lagrangian is the difference between them. 25. 41). Proof. A direct computation shows d 1 d gij (φ(t) )φ˙ i (t)φ˙ j (t) + U (φ(t) ) E(φ(t) ) = dt dt 2 1 ∂gij k i j ∂U s = φ˙ φ˙ φ˙ + gij φ¨ i φ˙ j + φ˙ .

18). In the case when M has a nonzero boundary, Hopf’s lemma becomes the uniqueness theorem for the Dirichlet problem. 7. Let M be a connected, compact manifold and f ∈ F(M) such that f = 0, f| ∂M = 0. on M, Then f ≡ 0. Proof. Integrate the expression div (f ∇f ) = −f f + |∇f |2 and use the divergence theorem div X dv = M with X = f ∇f . , df is one-to-one. Consider M as a Riemannian manifold with the induced metric by the immersion f , gij = f ∗ (δij ), where δij is the canonical metric on Rn .

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