By Thierry Aubin

This textbook for second-year graduate scholars is meant as an advent to differential geometry with critical emphasis on Riemannian geometry. bankruptcy I explains easy definitions and offers the proofs of the $64000 theorems of Whitney and Sard.

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

Chapter V specializes on Riemannian manifolds by way of deducing worldwide houses from neighborhood houses of curvature, the ultimate objective 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 money owed at the topic.

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7. Definition. Linear cotangent mapping ($*)p. Let P E Mn and Q = 4'(P). By duality, we define the linear cotangent mapping (V) p of TQ(W) into 7p(M) as follows: 7Q(W) 9 w , (4)p(w) E TA(M), (($*)p(w), X) _ (w, (4*)p(X)) for allX E Tp(M). 6) that d(f o P) p o X = (df) o ($) pX. 8. Proposition. 'l o ib* = (if o 4P)*. Let V be a third differentiable manifold and V a differentiable mapping of W into V. If f is a differentiable function in a neighbourhood of T(Q) 2. )-1. 9. Example. The tangent vector -2 to a differentiable curve -y(t) of M,, (y is a differentiable map of (a, b) C R into Mn).

28. But the Sard Theorem asserts that this result holds in general if the map f is Cl: the set of critical values is of measure zero. We are at the beginning of the course (in Chapter 1) and we have only seen some definitions and a few theorems, but nevertheless we were able to prove a very important theorem, the Whitney Theorem, which shows that a differentiable manifold, whose definition is abstract, is nothing else than a surface of dimension n in RP for p large enough. 30. Theorem (The Sard theorem).

A) [X, Y] = -[Y,X], and [X, Y] satisfies the Jacobi identity. 13) [X, [Y, Z]] + [Z, [X, Y]] + [Y, [Z, X]] = o. A real vector space L, endowed with a bilinear map L x L into L satisfying a) and 3), is called a Lie algebra. So the set of CO┬░ vector fields is a Lie algebra. A straightforward computation proves,3), and a) is obvious. 17. Definition. Projectable vector field. Let M and W be two differentiable manifolds and W a differentiable map of M into W. We have defined the linear tangent mapping, but in most cases it does not allow us to associate to a vector field X on M a vector field on W.

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