By R. Narasimhan

Chapter 1 provides theorems on differentiable features frequently utilized in differential topology, resembling the implicit functionality theorem, Sard's theorem and Whitney's approximation theorem.

The subsequent bankruptcy is an creation to genuine and intricate manifolds. It includes an exposition of the theory of Frobenius, the lemmata of Poincaré and Grothendieck with functions of Grothendieck's lemma to complicated research, the imbedding theorem of Whitney and Thom's transversality theorem.

Chapter three contains characterizations of linear differentiable operators, because of Peetre and Hormander. The inequalities of Garding and of Friedrichs on elliptic operators are proved and are used to turn out the regularity of vulnerable strategies of elliptic equations. The bankruptcy ends with the approximation theorem of Malgrange-Lax and its software to the facts of the Runge theorem on open Riemann surfaces as a result of Behnke and Stein.

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**Example text**

CLASSIFICATION: FREE CASE with a free symplectic action of T for which at least one, and hence every T -orbit is a (2n − 2)-dimensional symplectic submanifold of (M , σ ), and (M , σ ) is T equivariantly symplectomorphic to (M, σ), then the list of ingredients of (M , σ , T ) is equal to the list of ingredients of (M, σ, T ). Proof. Let Φ be a T -equivariant symplectomorphism from (M, σ) to (M , σ ). 2, the mapping Φ descends to a symplectomorphism Φ from the orbit space (M/T, ν) onto the orbit space (M /T, ν ).

3) ker(µ ) = (πIx )∗ (π1 (Ix , q0 )), 22 3. GLOBAL MODEL which in particular implies that the group (πIx )∗ (π1 (Ix , q0 )) is a normal subgroup of π1orb (Ix /S, q0 ), and we have a commutative diagram G π orb (Ix /S, q0 )/(πI )∗ (π1 (Ix , q0 )) π1orb (Ix /S, q0 ) 1 x UUUU UUUUµ U UUUU (fx )∗ UUUU UUB µ orb GS⊂T π (M/T, p0 ) 1 with the top arrow being the quotient map. Indeed, take an orbifold loop δ in Ix /S based at q0 . This implies that there exists a curve γ in Ix starting at q0 and such that δ = πIx ◦ γ.

The corresponding endpoints of the γ’s exhibit the integral manifold Ix as the image of an immersion from M/T into M , but this immersion is not necessarily injective. Recall that the monodromy homomorphism µ of Ω tells what the endpoint of γ is when δ is a loop. By replacing µ by the induced injective homomorphism µ from π1orb (M/T, π(x))/ker µ to T , we get an injective immersion. This procedure is equivalent to replacing the universal covering M/T by the covering M/T /ker µ with ﬁber π1orb (M/T, π(x))/ker µ.