Geometric Analysis by Peter Li

By Peter Li

The purpose of this graduate-level textual content is to equip the reader with the fundamental instruments and strategies wanted for learn in quite a few parts of geometric research. all through, the most topic is to give the interplay of partial differential equations and differential geometry. extra in particular, emphasis is put on how the habit of the recommendations of a PDE is plagued by the geometry of the underlying manifold and vice versa. For potency the writer usually restricts himself to the linear thought and just a rudimentary history in Riemannian geometry and partial differential equations is believed. Originating from the author's personal lectures, this booklet is a perfect advent for graduate scholars, in addition to an invaluable reference for specialists within the box.

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I p sgn(σ (I, I c )) ωim ∧ . . ∧ ωi p+1 . 24 Geometric Analysis By setting bk1 ... km− p = sgn(σ (I, I c )) ai1 ... i p with K = (k1 , . . , km− p ) = (i p+1 , . . , i m ) = I c , we can write β = bK ωK and dβ = b K , j ω j ∧ ω K = (dbk1 ... km− p + bk1 ... jθ ... km− p ω jθ kθ ) ∧ ω K for 1 ≤ θ ≤ m − p. On the other hand, we also have dβ = sgn(σ (I, I c )) dai1 ... i p ∧ ωim ∧ . . ∧ ωi p+1 + bk1 ... jθ... km− p ω jθ kθ ∧ ω K = sgn(σ (I, I c )) ai1 ... i p , j ω j ∧ ωim ∧ . . ∧ ωi p+1 − sgn(σ (I, I c )) ai1 ...

However, since one can view {∂/∂t, e1 , . . , em } as tangent vectors given by a coordinate system of (− , ) × M, we have the property that [T, dφt (ei )] = 0 for all 1 ≤ i ≤ m. 5) as 0 = ∇ei T, e j + ∇e j T, ei which is exactly the condition that T is a Killing vector field. 10 Let M be a compact manifold with nonpositive Ricci curvature. Then any Killing vector field on M must be parallel. Moreover, if there exists a point x ∈ M such that the Ricci curvature satisfies R(x) < 0, then there are 3 Bochner–Weitzenb¨ock formulas 31 no nontrivial Killing vector fields.

Now let us consider the case that N is an n-dimensional submanifold of M. Let {e1 , . . , em } be an adapted orthonormal frame field of M such that {e1 , . . , en } are orthonormal to N . We will now adopt the indexing convention 19 20 Geometric Analysis that 1 ≤ i, j, k ≤ n and n + 1 ≤ ν, μ ≤ m. The second fundamental form of N is given by ωνi = h iνj ω j . Relating the two notations, we have the formulas ωi j (ek ) = ∇ek ei , e j , Ri jkl = Rei e j el , ek , and − → h iνj = I I (ei , e j ), eν .

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