Walter Seaman

NC Verification of Up to 5 Axis Machining Processes Using Manifold Stratification

A numerically controlled machining verification method is developed based on a formulation for delineating the volume generated by the motion of a cutting tool on the workpiece (stock). Varieties and subvarieties that are subsets of some Eucledian space defined by the zeros of a finite number of analytic functions are computed and are characterized as closed form equations of surface patches of this volume. The motion of a cutter tool is modeled as a surface undergoing a sweep operation along another geometric entity. A topological space describing the swept volume will be built as a stratified space with corners. Singularities of the variety are loci of points where the Jacobian of the manifold has lower rank than maximal. It is shown that varieties appearing inside the manifold representing the removed material are due to a lower degree strata of the Jacobian. Some of the varieties are complicated and will be shown to be reducible because of their parametrization and are addressed. Benefits of this method are evident in its ability to depict the manifold and to compute a value for the volume.

On manifolds with nonnegative curvature on totally isotropic 2-planes

We prove that a compact orientable 2n-dimensional Riemannian manifold, with second Betti number nonzero, nonnegative curvature on totally isotropic 2-planes, and satisfying a positivity-type condition at one point, is necessarily Kahler, with second Betti number 1. Using the methods of Siu and Yau, we prove that if the positivity condition is satisfied at every point, then the manifold is biholomorphic to complex projective space

Harmonic two-forms in four dimensions

Conformal invariance of middle-dimensional harmonic forms is used to improve Katos inequality for four-manifolds. An application to positively curved four-manifolds is given. 0. INTRODUCTION The purpose of this paper is to prove the following: Theorem 1. Let (M4, g) be a four-dimensional Riemannian manifold. Let w be a harmonic two-form on (M, g). Then w satisfies the pointwise inequality: (0. 1 ) ~ ~~~~17(ol > 3ld l ,l,l Katos inequality [1, p. 130], states that if E is a Riemannian vector bundle with connection V over a Riemannian manifold M, then any smooth section s of E, satisfies the pointwise inequality: (0.2) IVS-2 > ldlsl 12 . Now by definition, if s(w) vanishes at p E M, then dlsl(dlwl) = 0 at p. Thus, (0.1) and (0.2) are automatically valid at such a point. At points where w does not vanish (0.1) can be thought of as a quantitative improvement of (0.2), for the case of harmonic two-forms on four-dimensional manifolds. As an application of the above theorem, we prove: Theorem 2. Let (M4, g) be a compact, connectedfour-dimensional Riemannian manifold whose sectional curvature K(g) satisfies 1 > K(g) > 3. If (0.3) 3 > 1/(3(1 + 3.2 14/51/2)1/2 + 1) . 1714 then M is definite. This theorem represents an improvement of results starting with [2] followed by [4, 7, 6]. The relevance of Theorems 1 and 2 stems from the following facts Received by the editors November 6, 1989 and, in revised form, July 27, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C20; Secondary 57N1 3.

Existence and uniqueness of algebraic curvature tensors with prescribed properties and an application to the sphere theorem

An existence and uniqueness theorem is proved for algebraic curvature tensors and then applied to yield a global geometric theorem for locally weakly quarter pinched Riemannian manifolds whose second Betti number is nonzero.

Two-forms on four-manifolds

We study the eigenvalues of the Weitzenbôck operator on a positively curved four-manifold, and give applications to its geometry and topology. The purpose of this paper is to prove the following results (see §2). Theorem 2. Let M be a Riemannian four-manifold. Suppose there is a point p g M where all sectional curvatures are positive. Then M admits at most one parallel two-form (up to constant multiples). The standard metrics on S4, CP2, and the (flat) four-torus T4, which have 0, 1, and 6 parallel two-forms, show that the hypothesis cannot be weakened to nonnegative and the conclusion is the best possible, at least within the framework of parallel forms. Also, this result fails if we replace sectional curvature with Ricci curvature as S2 X S2 shows (we thank Alfred Gray for pointing this out to us). Theorem 3. Let M be a compact positively curved four-manifold without boundary. If M admits a parallel two-form X, then the only harmonic two-forms on M are of the form cX, c g R. Corollary. A compact positively curved four-manifold M without boundary which has a parallel two-form X is necessarily a topological CP2. Proof of the Corollary. Assume first M is orientable. By Theorem 2, the only parallel two-forms on M are c ■ X, c g R. By Theorem 3, c ■ X is the only harmonic two-form on M. Hence H2(M; R) = R by Hodges theorem. Since any positively curved orientable even dimensional compact manifold is simply connected, we obtain H2(M;Z) = Z. Hence the intersection form of M is { + 1} or {-1}. By Freedmans work [2], M, as it has the intersection form class of CP2 or -CP2, and as M was assumed smooth, must be a topological CP2. If M is not orientable then its two-fold cover is CP2, as above. But x(CP2) = 3 = 2(M), a contradiction. Q.E.D. Both Theorems 2 and 3 follow from Theorem 1, a linear algebraic result which states that at any point of positive curvature, if the Weitzenbôck operator has kernel, then it has kernel dimension exactly one, and is positive definite on the orthogonal complement. Received by the editors January 16, 1986 and, in revised form, July 26, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C20. ©1987 American Mathematical Society 0002-9939/87

On Surfaces in R 4

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On four manifolds which are positively pinched

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Higgs bundles and holomorphic forms

We provide answers (Theorem C) to some questions concerning surfaces in R4 and maps into the quadric Q2 raised by D. Hoffman and R. Osserman. Let S be an oriented surface immersed in R4. The Gauss map of S is the map G of S into G(2, 4), the Grassmannian of oriented two-planes in R4, given by G( p) = TpS. G(2,4) can be identified with Q2, the complex quadric in CP3, and in turn Q2 is biholomorhic to CP x CP1. If we give CP3 the Fubini-Study metric of constant holomorphic sectional curvature 2, then the induced metric on Q2 is given by 21dw, 12/(1 + IW1 I)2 + 21dW2,2/(1 + IW21 ) where (w1, w2) are coordinates on C x C, viewed as local coordinates on CP x CP [1]. The metric 21dw12/(1 + Iwi2)2 is the metric on C induced by the map of C onto S2(1/ 4) c R3 given by w al1(J2iw), where a-1 is inverse stereographic projection (with the sphere sitting on the xy-plane). Thus, Q2 is isometric to S2(1/ x/) X S2(1/ vI). In particular, if z is a local conformal parameter on S, then any map G of S into Q2 splits into a pair of maps G(z) = (f1(z), f2(z)), where w, = fi(z) as above. Now define the following quantities on S for i = 1, 2: F,2:= T,(z)= (f1)2 wheref, O with the usual z and z derivative notation. The following results are from [1, 2]. THEOREM A. For the Gauss map G of an oriented surface S immersed in R4, write G = (f1(z), f2(z)) as above. Then we necessarily have (1) |1-I219 and (2) Im{T1+T2} O THEOREM B. Let SO be a simply connected Riemann surface (here and subsequently), let G = (f1(z), f2(z)) be some map of SO into Q2, and define Fi and Ti as before, where z is a conformal parameter on SO. (i) If F1 = F2 0, then G is the Gauss map of a minimal surface in R4, provided SO is not compact. Received by the editors April 19, 1984. 1980 Mathemnatics Subject Classification. Primary 53A05; Secondary 53A10. ?1985 American Mathematical Society 0002-9939/85

The Effects of Different Undergraduate Mathematics Courses on the Content Knowledge and Attitude towards Mathematics of Preservice Elementary Teachers.

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Orthogonally pinched curvature tensors and applications.

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