We handle the problem of finding a hypersurface family from a given asymptotic curve in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M1" height="12.1279pt" version="1.1" viewBox="-0.0498162 -12.0781 15.3921 12.1279" width="15.3921pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path id="g197-29" d="M712 0L497 285C582 312 628 378 628 466C628 609 518 662 390 662H70V0H236V271H310L505 0H712ZM491 604C551 577 584 532 584 464C584 401 550 345 491 321V604ZM447 320C428 317 411 315 392 315H236V618H397C412 618 432 613 447 610V320ZM626 44H528L363 271C392 271 422 270 451 275L626 44ZM192 44H114V618H192V44Z"/></g><g transform="matrix(.0095,0,0,-0.0095,10.12,-5.984)"><path id="g50-53" d="M462 177V227H365V632H320C217 496 116 350 21 208V177H284V109C284 43 280 38 190 31V0H451V31C369 38 365 43 365 108V177H462ZM284 227H88C155 336 218 430 282 519H284V227Z"/></g></svg>. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. We illustrate this method by presenting some examples.

Ergin Bayram

Emin Kasap

Hypersurface Family with a Common Isoasymptotic Curve

With the use of only the incidence axioms we prove and generalize Desargues&#x2019; two-triangle Theorem in three-dimensional projective space considering an arbitrary number of points on each one of the two distinct planes allowing corresponding points on the two planes to coincide and three points on any of the planes to be collinear. We provide three generalizations and we define the notions of a generalized line and a triangle-connected plane set of points.

Dimitrios Kodokostas

Proving and Generalizing Desargues&#x2019; Two-Triangle Theorem in 3-Dimensional Projective Space

The structure equations for a two-dimensional manifold are introduced and two results based on the Codazzi equations pertinent to the study of isometric surfaces are obtained from them. Important theorems pertaining to isometric surfaces are stated and a theorem due to Bonnet is obtained. A transformation for the connection forms is developed. It is proved that the angle of deformation must be harmonic, and that the differentials of many of the important variables generate a closed differential ideal. This implies that a coordinate system exists in which many of the variables satisfy particular ordinary differential equations, and these results can be used to characterize Bonnet surfaces.

Paul Bracken

An Intrinsic Characterization of Bonnet Surfaces Based on a Closed Differential Ideal

We classify some special Finsler metrics of constant flag curvature on a manifold of dimension <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.1625805pt" id="M2" height="8.76568pt" version="1.1" viewBox="0 -8.6031 28.8274 8.76568" width="28.8274pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"></path></g><g transform="matrix(.0135,0,0,-0.0135,10.586,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z"></path></g><g transform="matrix(.0135,0,0,-0.0135,22.324,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g></svg>.

Hongmei Zhu

A Local Classification of Some Special <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.53609pt" height="13.1824pt" version="1.1" viewBox="0 -9.64631 30.3047 13.1824" width="30.3047pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346" /></g><g transform="matrix(.0135,0,0,-0.0135,4.688,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="568" vert-adv-y="568" /></g><g transform="matrix(.0135,0,0,-0.0135,12.384,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228" /></g><g transform="matrix(.0135,0,0,-0.0135,17.744,0)"><path id="g113-224" d="M558 587C558 666 497 712 432 712C379 712 330 691 284 650C212 586 178 508 148 348L71 -65C49 -185 35 -229 23 -235L27 -261C49 -259 101 -251 124 -224C131 -216 138 -178 159 -24C171 66 197 200 227 356C264 550 295 668 393 668C443 668 479 632 479 575C479 516 446 458 383 418C361 404 344 398 318 397L296 350C395 338 460 281 460 192C460 110 411 40 300 40C258 40 215 55 192 69L181 51C191 18 222 -16 266 -16C308 -16 351 1 397 26C471 67 545 142 545 221C545 315 486 365 401 395C469 437 558 498 558 587Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="581" vert-adv-y="581" /></g><g transform="matrix(.0135,0,0,-0.0135,25.617,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346" /></g></svg>-Metrics of Constant Flag Curvature

This is a review paper of up-to-date research done on the existence of unique null curves, screen distributions, Levi-Civita connection, symmetric Ricci tensor, and scalar curvature for a large variety of lightlike submanifolds of semi-Riemannian (in particular, Lorentzian) manifolds, supported by examples and an extensive bibliography. We also propose some open problems.

K. L. Duggal

A Review on Unique Existence Theorems in Lightlike Geometry

We introduce the notion of paracomplex paracontact pseudo-Riemannian submersions from almost para-Hermitian manifolds onto almost paracontact metric manifolds. We discuss the transference of structures on total manifolds and base manifolds and provide some examples. We also obtain the integrability condition of horizontal distribution and investigate curvature properties under such submersions.

S. S. Shukla

Uma Shankar Verma

Paracomplex Paracontact Pseudo-Riemannian Submersions

In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition
on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence
results under perturbative and nonperturbative hypothesis, and with the help of some &#x201c;Morse inequalities at infinity&#x201d;, we provide multiplicity results for our problem.

Ridha Yacoub

Existence and Multiplicity Results for the Scalar Curvature Problem on the Half-Sphere <svg style="vertical-align:-3.30472pt" height="19.950001" version="1.1" viewBox="0 0 17.262501 19.950001" width="17.262501" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D54A" d="M525 174q0 -80 -75.5 -134.5t-195.5 -54.5q-57 0 -119 15q-60 17 -67 21q-8 23 -27 155l30 7q14 -57 49.5 -101.5t92.5 -44.5q47 0 73.5 28t26.5 76q0 51 -26.5 82.5t-83.5 66.5q-68 44 -93 73q-46 53 -46 124q0 75 68.5 126.5t182.5 51.5q84 0 157 -23l17 -138&#xA;q-5 -1 -14.5 -2.5t-14.5 -2.5q-10 30 -16 45t-18.5 36t-29.5 30.5t-41 9.5q-41 0 -62.5 -29t-21.5 -69q0 -51 26 -82.5t82 -65.5l42 -26.5t37 -28.5t36 -38.5t21 -46t10 -60.5zM482 174q0 89 -128 170q-126 75 -126 178q0 68 52 109q-68 -2 -103 -33.5t-35 -75.5&#xA;q0 -53 33 -97.5t87 -75.5q134 -78 134 -175q0 -96 -55 -142q63 16 102 52t39 90z" /></g><g transform="matrix(.012,-0,0,-.012,9.6,7.613)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5&#xA;q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g><g transform="matrix(.012,-0,0,-.012,9.6,19.863)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" /></g></svg>

The use of differential equations methods in the approach, treatment, and solution of problems in diverse areas of geometry, particularly in affine differential geometry is well known and prolific, where they have proven to be quite fruitful when it comes to the obtainment of definite results. It is perhaps lesser known that the same kind of those very same methods has been and is currently being used to treat developments in some specific areas of applied sciences, such as the theory of shells where, similarly, they can be proven to be quite effective as well. In this paper we precisely show that such is the case in two particular, related instances: the historic approach of the classical, Euclidean part of the theory pursued by Fritz John, in the past century, and the more recent expositions that we ourselves have dedicated to the affine counterpart of the theory.

Salvador Gigena

Daniel Abud

Mois&#xe9;s Binia

Geometrical and P.D.E. Methods in the Treatment  of the Theory of Shells: Comparing Euclidean and Affine Approaches

We prove that if <svg     style="vertical-align:-0.0pt;width:10.6125px;" id="M1" height="11.175" version="1.1" viewBox="0 0 10.6125 11.175" width="10.6125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D435" d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62&#xA;q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" /></g>

		
	
</svg> is a <svg     style="vertical-align:-0.1638pt;width:8.6750002px;" id="M2" height="12.4375" version="1.1" viewBox="0 0 8.6750002 12.4375" width="8.6750002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
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</svg>-positive holomorphic line bundle on a compact hyper-k&#xe4;hler manifold <svg     style="vertical-align:-0.0pt;width:17.475px;" id="M3" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28&#xA;q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g>

		
	
</svg>, then <svg     style="vertical-align:-2.3205pt;width:133.47501px;" id="M4" height="17.3375" version="1.1" viewBox="0 0 133.47501 17.3375" width="133.47501"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
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</svg> for <svg     style="vertical-align:-2.22495pt;width:91.862503px;" id="M5" height="15" version="1.1" viewBox="0 0 91.862503 15" width="91.862503"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
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</svg> with <svg     style="vertical-align:-3.50804pt;width:8.4750004px;" id="M6" height="12.1125" version="1.1" viewBox="0 0 8.4750004 12.1125" width="8.4750004"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45E"/></g>

		
	
</svg> a nonnegative integer. In a special case, <svg     style="vertical-align:-0.1638pt;width:36.262501px;" id="M7" height="12.4375" version="1.1" viewBox="0 0 36.262501 12.4375" width="36.262501"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
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</svg>, we recover a vanishing theorem of Verbitsky&#x2019;s with a little stronger assumption.

Qilin Yang

Vanishing Theorems on Compact Hyper-k&#xe4;hler Manifolds

There is a natural Moser type transformation along any curve in the moduli spaces of K&#xe4;hler metrics. In this paper we apply this transformation to give an explicit construction of the parallel transformation along a curve in the Mabuchi moduli space of K&#xe4;hler metrics. This is crucial in the proof of the equivalence between the existence of the K&#xe4;hler metrics with constant scalar curvature and the geodesic stability for the type II compact almost homogeneous manifolds of cohomogeneity one mentioned in (Guan 2013). We also explain a new description of the geodesics and prove a curvature property of the moduli space, called curvature symmetric, which makes it similar to some special symmetric spaces with nonpositive curvatures, although the spaces are usually not complete. Finally, we generalize our geodesic stability conjectures in (Guan 2003) and give several results on the Lie algebra structures related to the parallel transformations. In the last section, we generalize the Futaki obstruction of the K&#xe4;hler-Einstein metrics to the parallel vector fields of the invariant Mabuchi moduli space. We call the related stability the parallel stability. This includes the toric and cohomogeneity one cases as well as the spherical manifolds.

Daniel Guan

Moser Vector Fields and Geometry of the Mabuchi Moduli Space of K&#xe4;hler Metrics

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Geometry