جواب 1:

Aspointedoutbytheanswerbelow,spheresareexamplesofpositivelycurvedobjects,whilesaddles(theinsidepartofadonutforinstance)areexamplesofnegativelycurvedobjects.Averynicewaytounderstandcurvatureofmanifoldsistolookatgeodesicsonthem.Forsimplicity,letsjustconsidersurfaceslyinginEuclideanspacesuchasspheresandmultipleholeddonuts.Ageodesicisjustafancywordforashortestcurvebetweenanytwopoints(tobepreciseageodesicisonlylocallylengthminimizing,butletsforgetthatforamoment).Theconstrainisofcoursethatthecurveliescompletelyonthesurface.Soforexample,geodesicsonaspherearecontainedinequators.Thenitisafactthatpositivityofcurvatureisequivalenttothefactthattwogeodesicsleavingfromthesamepointtendtobendtowardseachotherwhiletheoppositehappensfornegativecurvature.Therearemanyequivalentwaysofrigorouslywritingthisout,eachofwhichcanbetakenasadefinitionofcurvature(thesocalledGausscurvatureofasurface).ThefollowingistheoneIlikethemost,sinceitrelatestotheveryfirsttheoremingeometrythatIstudiedsomanyyearsbacknamelythatthesumofangleofatriangleinaplaneis180degrees.Atriangleisapolygonwiththreestraightlinesides.Soonacurvedsurface,atrianglecanbedefinedasapolygonwiththreegeodesicsides.Thenintuitively,positivecurvaturewouldmeanfattertriangles.Letslookatanexample.Onasphereatriangleformedbyintersectingtheequatorwithtwolongitudes.Theanglecanbedefinedastheanglethatthetangentvectormakewitheachother.Thenthelongitudesmakerightanglewiththeequator.Soifwepicklongitudesthatarethemselvesatrightangles,wegetatriangleallofwhosesidesare90andhencethesumofanglesis270or3π/2whichisgreaterthanintheEuclideancase.Infact,anytrianglethatyoudrawwillhavesumofanglesgreaterthan[math]π[/math]!Thecurvatureisthenessentiallydefinedastheratioofthisdifferencebytheareaofthetriangle,asthetriangleshrinks.Forourexampleabove,thetriangleisanoctantofthesphere.SoifweassumethespherehasradiusR,thenthesurfaceareawillbe[math]\piR2/2[/math].Sotheratiois[math]1/R2[/math].Sincethesphereishighlyhomogenous,theratiowouldbethesamenomatterhowsmallatrianglewechoose.Sothecurvatureofasphereatanypointis[math]1/R2[/math]!Thisalsoagreeswithourintuition.Ahugeball,locallyisalmostflatandsomusthavetinycurvaturewhereasatinyballclearlyishighlycurved.As pointed out by the answer below, spheres are examples of positively curved objects, while saddles (the inside part of a donut for instance) are examples of negatively curved objects. A very nice way to understand curvature of manifolds is to look at geodesics on them. For simplicity, lets just consider surfaces lying in Euclidean space such as spheres and multiple holed donuts. A geodesic is just a fancy word for a shortest curve between any two points (to be precise a geodesic is only locally length minimizing, but lets forget that for a moment). The constrain is of course that the curve lies completely on the surface. So for example, geodesics on a sphere are contained in equators. Then it is a fact that positivity of curvature is equivalent to the fact that two geodesics leaving from the same point tend to bend towards each other while the opposite happens for negative curvature. There are many equivalent ways of rigorously writing this out, each of which can be taken as a definition of curvature (the so called Gauss curvature of a surface). The following is the one I like the most, since it relates to the very first theorem in geometry that I studied so many years back - namely that the sum of angle of a triangle in a plane is 180 degrees. A triangle is a polygon with three straightline sides. So on a curved surface, a triangle can be defined as a polygon with three geodesic sides. Then intuitively, positive curvature would mean fatter triangles. Lets look at an example. On a sphere a triangle formed by intersecting the equator with two longitudes. The angle can be defined as the angle that the tangent vector make with each other. Then the longitudes make right-angle with the equator. So if we pick longitudes that are themselves at right angles, we get a triangle all of whose sides are 90 and hence the sum of angles is 270 or 3\pi/2 which is greater than in the Euclidean case. In fact, any triangle that you draw will have sum of angles greater than [math]\pi[/math]! The curvature is then essentially defined as the ratio of this difference by the area of the triangle, as the triangle shrinks. For our example above, the triangle is an octant of the sphere. So if we assume the sphere has radius R, then the surface area will be [math]\piR^2/2[/math]. So the ratio is [math] 1/R^2 [/math]. Since the sphere is highly homogenous, the ratio would be the same no matter how small a triangle we choose. So the curvature of a sphere at any point is [math] 1/R^2 [/math]! This also agrees with our intuition. A huge ball, locally is almost flat and so must have tiny curvature whereas a tiny ball clearly is highly curved.

جواب 2:

یہ بدقسمتی حقیقت ہے: گھماؤ کی کوئی تعریف نہیں ہے۔ اس کے بجائے ، مختلف لیکن متعلقہ خیالات کا ایک پورا گروپ ہے جو سب کو گھماؤ کہتے ہیں۔ لہذا ، حقیقت میں یہ بتانے کے لئے کہ آپ کا مثبت اور منفی گھماؤ سے کیا مطلب ہے ، آپ کو پہلے کام کرنے کی ضرورت ہوگی کہ آپ کس گھماو کی تعریف کے ساتھ کام کر رہے ہیں۔

Forexample,forcircles,spheres,andthelike,weoftendefinethecurvaturetobe1/radiusn,where[math]n[/math]isthedimension(so[math]1[/math]foracircle,[math]2[/math]forasphere,andsoon)weintroduceapositiveandnegativesignifwestartthinkingaboutoriented[math]n[/math]spheres,wherewealsospecifywheretheinteriorofthe[math]n[/math]sphereis.Here,positivecurvaturemeansthattheinterioriswhereyouexpectittobe(aroundthecenterofthe[math]n[/math]sphere),andnegativecurvaturemeansthatitisswitchedwiththeexterior.For example, for circles, spheres, and the like, we often define the curvature to be 1/\text{radius}^n, where [math]n[/math] is the dimension (so [math]1[/math] for a circle, [math]2[/math] for a sphere, and so on)—we introduce a positive and negative sign if we start thinking about oriented [math]n[/math]-spheres, where we also specify where the interior of the [math]n[/math]-sphere is. Here, positive curvature means that the interior is where you expect it to be (around the center of the [math]n[/math]-sphere), and negative curvature means that it is switched with the exterior.

ThisisacompletelydifferentconventionforcurvaturethantheonecommonlyusedwhentalkingaboutRiemanniansurfaces,whichistheGaussiancurvature.TheGaussiancurvatureofasurfaceisdefinedinsuchawaythatitonlydependsonthesurface,andnotonhowitisembeddedintospace,sodistinctionslikeinterior/exteriorcannotfactorintoit.Indeed,theGaussiancurvatureofanyspherewillbe1/radius2.However,hyperboloidsdohavenegativeGaussiancurvatureintuitively,thisisbecauseatanygivenpoint,theybendinoppositedirectionsasyoumoveawayfromthispointindifferentdirections.This is a completely different convention for curvature than the one commonly used when talking about Riemannian surfaces, which is the Gaussian curvature. The Gaussian curvature of a surface is defined in such a way that it only depends on the surface, and not on how it is embedded into space, so distinctions like “interior/exterior” cannot factor into it. Indeed, the Gaussian curvature of any sphere will be 1/\text{radius}^2. However, hyperboloids do have negative Gaussian curvature—intuitively, this is because at any given point, they bend in opposite directions as you move away from this point in different directions.

گھماؤ کے دوسرے تصورات کی بھی بہتات ہیں — میں دلچسپی رکھنے والے قاری کو ویکیپیڈیا کے صفحے پر حوالہ دیتا ہوں ، جو دراصل بہت جامع ہے۔

جواب 3:

خلائی وقت میں ، مثبت گھماؤ بڑے پیمانے پر اور توانائی کی وجہ سے ہوتا ہے ، جس سے کشش ثقل پیدا ہوتا ہے۔ منفی گھماؤ منفی توانائی کی وجہ سے ہوتا ہے جس کی کائنات میں بہت کم مقدار ہوتی ہے (ملاحظہ کریں "کاسمیر اثر"۔) شعبوں میں مثبت گھماؤ ہوتا ہے ، زینوں میں منفی گھماؤ ہوتا ہے۔ منفی توانائی ہمیں مخالف معنی میں خلا کو موڑنے کے قابل بنائے گی جو معاملہ کرتا ہے ، اس سے معاہدہ کرسکتا ہے ، اس کے ساتھ کھلے کیڑے کی چھڑیاں لگائیں گے (دیکھیں کیپ تھورن کا کام اور دوسرے دیکھیں۔) اس سے نفرت انگیز قوت پیدا ہوگی۔

جواب 4:

خلائی وقت میں ، مثبت گھماؤ بڑے پیمانے پر اور توانائی کی وجہ سے ہوتا ہے ، جس سے کشش ثقل پیدا ہوتا ہے۔ منفی گھماؤ منفی توانائی کی وجہ سے ہوتا ہے جس کی کائنات میں بہت کم مقدار ہوتی ہے (ملاحظہ کریں "کاسمیر اثر"۔) شعبوں میں مثبت گھماؤ ہوتا ہے ، زینوں میں منفی گھماؤ ہوتا ہے۔ منفی توانائی ہمیں مخالف معنی میں خلا کو موڑنے کے قابل بنائے گی جو معاملہ کرتا ہے ، اس سے معاہدہ کرسکتا ہے ، اس کے ساتھ کھلے کیڑے کی چھڑیاں لگائیں گے (دیکھیں کیپ تھورن کا کام اور دوسرے دیکھیں۔) اس سے نفرت انگیز قوت پیدا ہوگی۔