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Stanford MATH 396 - Operations with pseudo-Riemannian metrics

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Math 396. Operations with pseudo-Riemannian metricsWe begin with some preliminary motivation. Let (V, h·, ·i) be a finite-dimensional vector spaceendowed with a non-degenerate symmetric bilinear form (perhaps not p ositive-definite). In linearalgebra, we have seen how to carry out several operations in the presence of this structure. Forexample, if V0is a subspace of V then we can form its orthogonal complement V0⊥(the subspace ofv ∈ V such that hv, v0i = 0 for all v0∈ V ), and this is also a complementary subspace: V0⊕V0⊥→ Vis an isomorphism. We have also s een in §2 of the old handout on orientations how to endow V⊗n,Symn(V ), and ∧n(V ) with non-degenerate symmetric bilinear forms, as well as V∨(the “dual”bilinear form), and that using the one on V∨likewise gave rise to non-degenerate symmetric bilinearforms on (V∨)⊗n, Symn(V∨), and ∧n(V∨) such that the natural isomorphisms(V∨)⊗n' (V⊗n)∨, Symn(V∨) ' (SymnV )∨, ∧n(V∨) ' (∧nV )∨are compatible with the bilinear forms built on the left via the dual form on V∨and the bilinearforms on the right that are dual to the ones built above on the tensor, symmetric, and exteriorpowers of V .Rather concretely, on V⊗n, Symn(V ), and ∧nV the induced symmetric bilinear forms werecharacterized uniquely by the conditionshv1⊗ · · · ⊗ vn, v01⊗ · · · ⊗ v0niV⊗n=Yihvi, v0ii, hv1· · · vn, v01· · · v0niSymnV=Xσ∈SnnYi=1hvi, v0σ(i)i,hv1∧ · · · ∧ vn, v01∧ · · · ∧ v0ni∧nV=Xσ∈Snsgn(σ)nYi=1hvi, v0σ(i)i = det(hvi, v0ji),and on V∨the dual bilinear form was defined by h`, `0iV∨= hv, v0i where ` = hv, ·i and `0= hv0, ·i.For example, if {vi} is a basis of V then the element of (∧nV )∨⊗ ∧n(V )∨' ∧n(V∨) ⊗ ∧n(V∨) thatcorresponds to h·, ·i∧n(V )isXj1<···<jn,j01<···<j0ndet(hvjr, vj0si(r,s))(v∗j1∧ · · · ∧ v∗jn) ⊗ (v∗j01∧ · · · ∧ v∗j0n)because of two facts: (i) det(hvjr, vj0si) = hvj1∧ . . . vjn, vj01∧ · · · ∧ vj0ni∧n(V ), and (ii) under theisomorphism (∧nV )∨' ∧n(V∨) the basis of ∧n(V∨) given by wedge products of v∗j’s is the dualbasis to the basis of ∧n(V ) given by wedge products of vj’s.In Corollary 2.3 of the orientation handout, we saw explicitly how to use an orthonormal basis{ei} of V (i.e., hei, eji = 0 for i 6= j and hei, eii = ±1 for all i) to build orthonormal bases of all ofthese auxiliary spaces, and so in particular when h·, ·i is positive-definite it follows that the inducedbilinear forms on all of these auxiliary s paces are also positive-definite. That is, if V is an innerproduct space then all of the spacesV⊗n, Symn(V ), ∧n(V ), V∨, (V∨)⊗n, Symn(V∨), ∧n(V∨)are naturally inner product spaces, and from an orthonormal basis of V we can build orthonormalbases on all of these other s paces. For example, the dual basis to an orthonormal basis is orthonor-mal, and tensor and wedge products of members of an orthonormal basis are orthonormal. Recallthat for symmetric powers there were certain factorials that intervened in the formation of an or-thonormal basis from one for V , ultimately due to the intervention of factorials in the descriptionof the duality be tween Symn(V ) and Symn(V∨) in terms of a basis of V and its dual basis in V∨.Our aim here is to show how all of these matters can be carried out for vector bundles endowedwith a pseudo-Riemannian metric. In a sense, the real work all took place in linear algebra: all we12have to do here is chase some “universal” fibral formulas. The universal nature of the constructionsin linear algebra is “why” everything will carry over to vector bundles.The most important case of all (for geometric applications) is to use a Riemannian metric on thetangent bundle E = T M to a Riem annian manifold with corners M to build Riemannian metrics onthe bundles ∧n(E∨) = ΩnMof smooth differential forms, esp ec ially the line bundle det(T∗M) of top-degree differential forms. Indeed, with respect to the induced Riemannian metric on this line bundleand an “orientation” on the manifold M (to be discussed later), there will be a distinguished top-degree form that is just the “volume form” on fibers, as defined in the old handout on orientations.The reason for the name volume form in that old handout was exactly this application, where theyglue together to give a special top-degree differential form on an oriente d Riemannian manifold.In terms of the theory of integration to be developed for top-degree differential forms on arbitraryoriented manifolds, integration of this volume form will provide the definition for volume of opensubsets of an oriented Riemannian manifold. Applied in the case of surfaces in R3given the inducedmetric tensor (from the flat one on R3), this procedure will recover all of the classical formulasfor surface area of open subsets of various classes of (oriented) surfaces with boundary (such assurfaces of revolution, or the parametric “z = f(x, y)” sort, etc.). In particular, we will arrive atthe striking fact that the induced metric tensor on an embedded surface in R3is all we need todetermine a satisfactory (and physically reas onable) theory of area on the surface.Even though all that follows will largely be used in this course only for the tangent bundle andbundles derived from it by the standard tensor operations and duality, in any deeper study ofdifferential geometry one quickly finds it necessary to consider many other kinds of vector bundlesand so both practical applications as well as conceptual clarity inspire the decision to develop therest of this handout for arbitrary vector bundles with pseudo-Riemannian metric, and not just forthe tangent bundle (where it may b e too tempting to phrase everything in the language of vectorfields and differential forms, which is rather irrelevant to the “vector bundle linear algebra” beingcarried out).1. Basic construction and examplesRecall that a finite-dimensional quadratic space (V, q) over R is c alled non-degenerate when theassociated symmetric bilinear form Bq(v, v0) = q(v + v0) − q(v) − q(v0) sets up a perfect pairingbetween V and itself: v 7→ Bq(v, ·) = Bq(·, v) is a linear isomorphism from V to V∨. If dim V = n,recall that we classified all such pairs (V, q) up to isomorphism by means of the signature (r, s),where r, s ≥ 0 and in some system of linear coordinates q =Pri=1x2i−Psj=1x2r+j(so


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