By Thierry Aubin

This textbook for second-year graduate scholars is meant as an advent to differential geometry with crucial emphasis on Riemannian geometry. bankruptcy I explains simple definitions and provides the proofs of the real theorems of Whitney and Sard.

Chapter II bargains with vector fields and differential kinds. bankruptcy III addresses integration of vector fields and $p$-plane fields. bankruptcy IV develops the inspiration of connection on a Riemannian manifold regarded as a way to outline parallel shipping at the manifold. the writer additionally discusses comparable notions of torsion and curvature, and offers a operating wisdom of the covariant spinoff.

Chapter V specializes on Riemannian manifolds by means of deducing international houses from neighborhood houses of curvature, the ultimate target being to figure out the manifold thoroughly. bankruptcy VI explores a few difficulties in PDEs steered by means of the geometry of manifolds.

The writer is famous for his major contributions to the sector of geometry and PDEs--particularly for his paintings at the Yamabe problem--and for his expository bills at the topic.

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**Additional resources for A Course in Differential Geometry (Graduate Studies in Mathematics)**

**Sample text**

17. Definition. Let M, M be two Ck differentiable manifolds. M is called a covering manifold of M if there exists a differentiable mapping II (called a projection) of M onto M. such that for every P E M: i) II-1(P) is a discrete space F, and ii) there exists a neighbourhood 11 of P such that 11' (Q) is diffeomor- phic to flxF. Each point P E 11-1(P) has a neighbourhood f2' C M such that the restriction II' of II to SY is a diffeomorphism of IT onto Q. The map II is a 2-sheeted covering if F consists of two points.

Exhibit two real numbers m and M such that f (x)em < a < f (x)eM for all x E R. Then consider the sequence of functions defined by induction as follows: cpo = m and, for k > 0, pk is the solution in E2 of the equation (5) f Wk + Vk cotan x - (x)elk-' -- a - Wk-1, where p > 0 is a real number. Prove that Wr > 00. c) If p is chosen large enough, establish that the sequence {Wpk} is increasing and bounded by M. d) Prove that {cpk} is a Cauchy sequence in C. Deduce that equation (4) has a unique solution in E2.

What is its dimension? Hint. Apply in $; a theorem of the course. For the compactness look at the proof of compactness of P2. 46. Exercise. Let Mp and W. be two C°° differentiable manifolds of dimension p and n respectively, and let f be a CO0 map from M. into W. We say that f is a subirnnersion at x E Mp if there exist a neighbourhood n C Mp of x, a neighbourhood 0 C W of f (x), a C°°° manifold V, and two C°° maps g and h such that g is a submersion of fE into V and h is an immersion of V into Wn with f lo = h o g.