The Knowledge Frontier

LissafiKafin bugawaNazariyyaKaratun minti 6

ƘWALLO MAI SANTSI DA KE ZAGAYE A WURI ƊAYA KAWAI

Ƙwallo yana lanƙwasa iri ɗaya a kowace hanya. Gefen ƙwallon rugby ba haka yake ba: yana lanƙwasa da ƙarfi a kewayen ƙugunsa, kuma a hankali kawai a tsawonsa. A kowane wuri na saman, masana ilimin siffofi (geometers) suna lura da lanƙwasa mafi ƙarfi da mafi rauni, waɗanda hanyoyinsu a koyaushe suke a kusurwa madaidaiciya (right angle) da juna; waɗannan su ne manyan lanƙwasa biyu (principal curvatures). Wurin da suke daidai, inda saman yake lanƙwasa iri ɗaya a dukan hanyoyi, ana kiransa umbilic. A kan cikakken ƙwallo, kowane wuri umbilic ne.

Ƙirga juye-juye a kewayen umbilic

Nesa da umbilics, hanyoyin lanƙwasa mafi ƙarfi da mafi rauni suna zana iyalai biyu na layuka a kan saman, waɗanda ke ratsa juna a kusurwa madaidaiciya a ko’ina. Ka zagaya sau ɗaya a kewayen umbilic keɓaɓɓe ta wata ƙaramar hanya mai zagaye (loop), kuma ka bi ɗaya daga cikin waɗannan layuka: yana juyawa. Adadin juye-juyen da yake yi shi ne index na umbilic ɗin. Saboda layi ba shi da kibiya, zai iya komawa kansa bayan rabin juyi, don haka index zai iya zama rabin cikakkiyar lamba kamar 3/2.

Hasashe biyu, kamar yadda takardar ta bayyana su:

  • Hasashen Loewner: index na umbilic keɓaɓɓe bai wuce 1 ba.
  • Hasashen Carathéodory: rufaffen saman mai kumbura waje kuma mai daidaito (closed convex regular surface: saman mai santsi mara gefe, wanda ke kumbura waje a ko’ina, kamar ƙwai) a cikin sararin fuska uku na yau da kullum yana da aƙalla umbilics biyu.

Su biyun suna da alaƙa. A kan rufaffen saman mai kumbura waje wanda umbilics ɗinsa keɓaɓɓu ne, jimillar index ɗinsu a koyaushe 2 ce. Idan babu wanda zai iya wuce 1, za a buƙaci aƙalla umbilics biyu: hasashen Loewner zai tabbatar da na Carathéodory. Umbilic guda ɗaya tilo zai ɗauki dukan 2 shi kaɗai, kuma ya karya duka biyun.

An sanar a watan Agusta, an sake gina shi da sines

Misalan farko da suka saɓa ba daga wannan takarda suka fito ba. A cewar gabatarwarta, L. Alpöge ya sanar da misalan da suka saɓa wa duka hasashen biyu ga sama masu santsi, “an samar da su da taimakon AI” (developed with the aid of AI), a wani rubutu a X na ranar 19 ga Agusta 2026. Ginin gida (local construction) na Alpöge ya riga ya kai index na 1 + k/2 ga kowace cikakkiyar lamba tabbatacciya (positive whole number) k, kuma Alpöge ya ba da rahoton rufaffen saman mai kumbura waje sosai (strictly convex) da umbilic ɗaya daidai, mai index 2.

Naoya Ando, na Kumamoto University, da Masaaki Umehara, na Institute of Science Tokyo, sun yi niyyar ba da “hanyar gini mafi sauƙi ta wani nau’in ginin Alpöge” (a simpler construction through a variant of Alpöge’s construction). Aikin lissafi (function) na farko na Alpöge yana haɗa cosines uku da masu ninkawa (coefficients) −1/4, 3/10 da −1/32, a cikin ayyukan lissafi masu ɗauke da ma’auni masu dorewa (constants) kamar 10¹⁰ da 100. “Irinsu” (seed) shi ne:

F(x, y) = a(x) − a(y) + sin x sin y, inda a(t) = sin⁴t + 3 sin²t

Ka yi tunanin aikin lissafi na masu canzawa (variables) biyu a matsayin shimfiɗar ƙasa mai tuddai da kwazazzabai. Irin yana buƙatar siffa ɗaya kawai: babu inda wannan shimfiɗa za ta lanƙwasa iri ɗaya a dukan hanyoyi. A kalmomin fasaha, trace-free Hessian ɗinsa, wato teburin abubuwan samowa na biyu (second derivatives) da aka cire masa ɓangaren “iri ɗaya a kowace hanya”, ba ya taɓa zama sifili. Ga wannan iri, tabbatarwar tana ɗaukar layuka uku. An tattara siffar a cikin lamba ɗaya mai sarƙaƙiya (complex number) — ma’auratan lambobi na yau da kullum, ɓangare “na gaske” (real) da ɓangare “na hasashe” (imaginary), wadda ke zama sifili ne kawai idan duka biyun sifili ne: 12 − 16(sin⁴x + sin⁴y) + 2i cos x cos y. Ɓangarenta na hasashe yana zama sifili ne kawai inda cosine yake sifili; a can sine yana daidai da ±1, don haka ɓangarenta na gaske bai wuce −4 ba. Don haka ba ta taɓa zama sifili.

Marubutan sun bayyana cewa sun yi amfani da ChatGPT don taimakawa wajen neman ayyukan lissafi masu yiwuwa da kuma gyara rubutun, kuma sun tabbatar da kowace hujja da kansu.

Naɗe tsari marar ƙarshe zuwa wuri ɗaya

Irin yana maimaituwa har abada a kan dukan fili (plane). Marubutan sun matse shi gaba ɗaya cikin ƙaramin faifai (disc): H_k(z) = exp(−|z|^(−k/4)) F(z^(−k/2)), inda z wani wuri ne na filin da aka rubuta a matsayin lamba ɗaya mai sarƙaƙiya, k kuma cikakkiyar lamba tabbatacciya da marubutan ke da ’yancin zaɓa. Musanyawar tana tura can nesa na filin zuwa tsakiya, don haka tsarin sine yana taruwa a cikin zobba masu ƙara matsewa a kewayensa. Ɓangaren exponential yana murƙushe komai: a tsakiya, aikin lissafin da dukan derivatives ɗinsa sifili ne. Masana lissafi suna kiran irin wannan aiki flat (lebur).

A kewayen wannan tsakiya, hanyoyin lanƙwasa mafi ƙarfi na wannan shimfiɗa suna juyawa sau 1 + k/2: 3/2 ga k = 1, 2 ga k = 2, 5/2 ga k = 3, ba tare da iyakar sama ba (Theorem 1.3). Tare da Fujiyama, waɗannan marubuta biyu sun riga sun kai index masu girma yadda ake so a baya, da misalai waɗanda C¹ ne kawai, wato ana iya samo derivative ɗinsu sau ɗaya kawai, a wurin umbilic; waɗannan sababbi suna da santsi.

Don mayar da shimfiɗa saman, marubutan suna amfani da aikin tallafi (support function): ga kowace hanya, yana ba da nisa daga wata tsakiya zuwa lebur ɗin fili da ke taɓa saman a wannan gefen, kamar rula da aka danna a kan tsakuwa. Aikin tallafi mai dorewa (constant) yana bayyana siffar ƙwallo (sphere). Ƙara ƙaramin ninki na shimfiɗar a kan wannan ma’auni mai dorewa yana ba da saman wanda umbilics da index ɗinsa suka yi daidai da na shimfiɗar. Wannan alaƙa ce ta gargajiya; takardar tana tabbatar da ɓangaren da take buƙata kawai don rufaffen saman kuma tana ambaton sauran. Sakamako: ƙananan guntayen saman mai santsi masu ɗauke da umbilics na index 1 + k/2 ga kowane k, don haka index ba shi da rufi. Haka yake ga wani rukuni na sama a sararin Lorentz–Minkowski, inda, a cikin dabarar Pythagoras ta nisa, ake cire murabba’in ɗaya daga cikin bambance-bambancen ma’auni (coordinates) uku maimakon a ƙara shi.

Umbilic guda ɗaya tilo yana a sandar arewa

Don rufaffen saman, ana naɗe filin marar ƙarshe na irin a kewayen siffar ƙwallo, kamar zanen taswira (map projection) da aka gudanar a baya: yana rufe kowane wuri sai sandar arewa, inda dukan can nesansa ke haɗuwa. Ana rage ƙarfin irin ta wata hanya dabam, don ya dusashe da sauri fiye da kowane ƙarfi (power) na nisa daga tsakiyar wannan fili, kuma ana ƙara ƙaramin ninki ε nasa a kan aikin tallafi na siffar ƙwallon. Idan ε ya isa ƙanƙanta, sakamakon shi ne rufaffen saman, kusan zagaye, mai kumbura waje sosai (strictly convex): yana kumbura waje a ko’ina. A kowane wuri sai ɗaya, siffar irin tana hana umbilic. Banbancin shi ne sandar arewa, N = (0, 0, 1): a can, aikin tallafi ya yi daidai da na siffar ƙwallon a kowane derivative, don haka N umbilic ne, kuma shi kaɗai ne (Theorem 2.2). Tun da dole jimillar index ta zama 2, index ɗinsa 2 ne. Umbilic ɗaya: hasashen Carathéodory ya faɗi. Index 2, fiye da 1: haka ma na Loewner.

Mai santsi, amma ba analytic ba

Kowane saman a cikin takardar ana iya samo derivatives ɗinsa har abada (infinitely differentiable), wato “mai santsi” (smooth). Marubutan sun fayyace cewa samansu a sararin yau da kullum ba real analytic ba ne, wani nau’in santsi mafi tsauri wanda a cikinsa, kusa da kowane wuri, ana ba da siffar daidai ta jimillar ƙarfafa (powers) marar iyaka. Takardar ba ta ce komai ba game da ko sama masu analytic za su iya samun umbilic guda ɗaya tilo. Ta bayyana kanta a matsayin wadda ta isa kanta (self-contained) ga umbilic na index 2 da misalin da ya saɓa wa hasashen Carathéodory. Kuma ba ta ƙunshi hoton saman ba: ma’auni masu dorewa biyu da ke ba shi siffa, δ da ε, ana buƙatar su ne kawai su zama “ƙanana sosai” (sufficiently small), kuma ba a ba da wata ƙima ba.