DIFFERENTIAL GEOMETRY

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1 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Differential Geometry - Taashi Saai DIFFERENTIAL GEOMETRY Taashi Saai Deartment of Alied Mathematis, Oayama University of Siene, Jaan Keywords: urve, urvature, torsion, surfae, Gaussian urvature, mean urvature, geodesi, minimal surfae, Gauss-Bonnet theorem, manifold, tangent sae, vetor field, vetor bundle, onnetion, tensor field, differential form, Riemannian metri, urvature tensor, Lalaian, harmoni ma Contents. Curves in Eulidean Plane and Eulidean Sae. Surfaes in Eulidean Sae. Differentiable Manifolds 4. Tensor Fields and Differential Forms 5. Riemannian Manifolds 6. Geometri Strutures on Manifolds 7. Variational Methods and PDE Glossary Bibliograhy Bibliograhy Seth Summary After the introdution of oordinates, it beame ossible to treat figures in lane and sae by analytial methods, and alulus has been the main means for the study of urved figures. For examle, one attahes the tangent line to a urve at eah oint. One sees how tangent lines hange with oints of the urve and gets an invariant alled the urvature. C. F. Gauss, with whom differential geometry really began, systematially studied intrinsi geometry of surfaes in Eulidean sae. Surfae theory of Gauss with the disovery of non-eulidean geometry motivated B. Riemann to introdue the onet of manifold that oened a huge world of diverse geometries. Current differential geometry mainly deals with the various geometri strutures on manifolds and their relation to toologial and differential strutures of manifolds. Results in linear algebra (Matries, Vetors, Determinants and Linear Algebra) and Eulidean geometry (Basi Notions of Geometry and Eulidean Geometry) are assumed to be nown as aids in enhaning the understanding this hater.. Curves in Eulidean Plane and Eulidean Sae Plane urves A urve in Eulidean lane with orthogonal oordinates ( x, x) is regarded as the lous of a moving oint with time and given by a arametri reresentation x () t = ( x () t, x ()) t, a t b. ()

2 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Differential Geometry - Taashi Saai It is assumed that funtions x () t, x () t are of lass C ( ) and the tangent vetor x () t = ( x () t, x ()) t to is nonzero at eah t. Then the tangent line to at x ( t0 ) is given by t x( t0) + tx ( t0) in vetor notation. The ar length st () and the length L () of are defined as t x t x b x a a a s() t = () t dt= () t + () t dt ( a t b), L( ) = x () t dt. () Indeed, the length L () doesn t deend on arameterizations of a urve. Sine s= s() t is a stritly inreasing funtion, one may tae the inverse funtion t = t() s and gets a new arametri reresentation s x() s = x (()) t s of by ar length s for whih x () s holds. Setting e () s = x () s and e () s = ( x () s, x ()) s, a unit normal vetor to given by rotating e () s through 90 ounterlowise, one obtains a (ositive) orthonormal basis { e( s), e ( s)} alled the Frenet frame of at eah x ( s). Curvature Let : x= x ()0 s, s L be arameterized by ar length. Then the aeleration vetor x () s = ( x() s, x()) s of is orthogonal to x () s, and one may write x() s = e () s =κ () s e() s, where κ()( s = κ ()) s = x() s, e () s = x () s x() s x () s x() s is alled the urvature of at x ( s). Setting ρ( s) = / κ ( s) if κ ( s) 0, the irle entered at x() s + ρ() s e () s (enter of urvature) of radius ρ () s is tangent to of seond order at x () s. The enters of urvature of form a urve alled the evolute of. For a urve : x= x () t arameterized by t, one obtains { x } { } x x x x x κ () t = () t () t () t () t () t + () t. () Here is an examle: The lous of a fixed oint on the irle of radius a rolling on x - axis is alled the yloid and given by x () t = a( t sin t), x () t = a( os)0 t, t π. Its ar length and urvature are resetively st ( ) = a{ os( t/ )} and κ () t = {4 / asin( t/ )}. The evolute of a yloid defined for < t < is again a yloid that is ongruent to. What is the meaning of urvature? It aears in the Frenet formula: x () s = e () s, e () s = κ () s e () s, e () s = κ () s e () s (4) that ontrols the loal behavior of the Frenet frame and the urve itself. Now an angle θ () s R between the unit tangent vetor x ( s) to and a fixed unit vetor may be defined so that s θ ( s) is of lass C and one has x ( s) = (os( θ ( s) + α), sin( θ( s) + α)). Then κ ( s) = θ ( s) holds, i.e. urvature is an intrinsi invariant of a urve. Indeed, urvature determines the urve: Suose a C -

3 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Differential Geometry - Taashi Saai funtion κ ( s) is given on [0, L]. Then there exists a unique C -urve : = ( s) x x in arameterized by ar length with κ ( s ) κ ( s ), u to arallel translations and rotations of. In the standard ase of κ ( s) 0 (res. κ ( s) 0 ), is a (art of a) straight line (res. irle of radius ). A urve given by the equation κ ( s) = s/ a (a ; a nonzero onstant) is alled a lothoid that desribes the trajetory of a ar running with unit seed and an inreasing aeleration of the onstant rate, and alied to the design of highways. Figure. Cyloid and its evolute (left); lothoid (right) Let : [0, L] be a losed urve, i.e. ( ) (0) ( x = x ) ( L) hold for -th derivatives L ( 0 ). The integral κ () sdsis alled the total urvature of, and the rotation 0 number of is given by (/ π) κ( sds ) that turned out an integer. For a simle (i.e. 0 L without self-intersetion oints) losed urve, the rotation number is equal to ±, and two losed urves are deformed to eah other (by regular homotoy) if and only if they have the same rotation number. A simle losed urve admits at least four verties at whih the derivative κ () s vanishes. Sae urves { x x x xi } A urve in Eulidean sae ( ) =,, is given by x () t = ( x() t, x() t, x()) t, a t b (5) using a arametri reresentation. It is assumed that xi ( t ) ( i ) are of lass C ( ) and the aeleration vetor x () t = ( x() t, x() t, x()) t is linearly indeendent of the tangent vetor x () t = ( x () t, x () t, x ()) t. Ar length s= s() t is defined by () and one gets the arameterization of by ar length. Then e () s = x () s is a unit vetor and x () s is orthogonal to x ( s). Now the urvature κ ()( s = κ ()) s of a sae urve is

4 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Differential Geometry - Taashi Saai defined as x () s that is assumed to be ositive. Set e () s = x () s /κ () s and onsider the vetor rodute() s = e() s e () s. One obtains a (ositive) orthonormal basis { e i( s)} i alled the Frenet frame at eah oint x ( s) of the urve. Then the following Frenet-Serret formula e () s = κ() s e () s e () s = κ() s e() s + τ() s e() s e () s = τ () s e () s (6) holds, where τ()( s = τ ()) s = e () s, e () s = det(() x s, x() s, x()) s x() s is alled the torsion of (det means the determinant of -matrix formed by the omonents of three vetors). The urvature and the torsion of a urve x= x ( t) arameterized by t are given by κ() t = x () t x() t x () t, τ() t = det( x () t, x() t, x()) t x () t x() t. (7) One κ() s > 0, τ()(0 s s L) are given, there exists a unique urve arameterized by ar length with κ ( s) = κ( s), τ ( s) = τ( s), u to arallel translations and rotations of. is a lane urve if and only if its torsion vanishes everywhere, and a urve with onstant urvature κ ( s) a> 0 and onstant torsion τ ( s) b 0 is ongruent to a regular helix given by x ( s) ( a b ) ( aos( a b s) asin( a b s) b a b s) = + +, +, +. For the total urvature κ = κ() sdsof a losed urve : x= x ( s)(0 s L), κ π holds. Moreover if is a not, κ > 4π holds.. Surfaes in Eulidean Sae 0 L Fundamental forms and urvature In nature one sees various urved surfaes and nowadays urved surfaes are ut use to designs, e.g. for ars. In differential geometry, one treats arameterized surfae S given by a ma x: D ( u, u ) ( u, u ) = ( x ( u, u ), x ( u, u ), x ( u, u )) x (8) from a domain D of uu -lane into, where xi ( u, u) ( i ) are funtions of lass C ( ). For examle, the grah of a funtion x = f( x, x ) is exressed as ( x, x) ( x, x, f( x, x)). The shere x + x + x = r of radius r is given by x = rosu osu, x = rosu sin u, x = rsin u ; π/ < u < π/, 0 u π. To 4

5 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Differential Geometry - Taashi Saai guarantee that S is a -dimensional figure, one taes urves u x( u, u), u x( u, u) on S fixing u, u resetively and assumes that the tangent vetors to these arameter urves x u i x x x ( u, u) = ( u, u), ( u, u), ( u, u),( i =, ) ui ui ui are linearly indeendent and san the tangent lane Namely, the vetor rodut u u x ( ). TS to S at eah = u, u x ( u, u ) x ( u, u ) 0 everywhere, and one may tae e( u, u ) = x ( u, u ) x ( u, u ) x ( u, u ) x ( u, u ), (9) u u u u a unit normal vetor to S. x : D is assumed to be injetive on D. Now the salar rodut is indued on eah TS. In terms of the first fundamental quantities g = x, x ( i, j =, ; g = g ) (0) ui u j it is given by gab i j i j ξ, ξ = for ξ, = = a ixu, ξ i i = b i ix u T i S. One may = = onsider the norm of a vetor and the angle between vetors in TS. For examle, if a urve on S is given by [ ab, ] t x( u( t), u( t)), where t ( u( t), u( t)) is a C - urve in D, one has x () t = u () i t xu T i i ( u( t), u( t)) S = x. Then x () t = g u i() t u j() t holds and the length L () of is given by (). The salar rodut on TS doesn t deend on arameterizations of the surfae S, and a quadrati form ds = g i j duidu, = j on TS is alled the first fundamental form. The area of S is given by D () A( S) = ds where ds = det( g ) du du. The Gauss ma G: S S is defined by assigning the unit normal vetor e ( u, u) to eah oint = x ( u, u) of S, where S denotes the unit shere in. To see how the unit normal vetor e behaves on S, the seond fundamental quantities are introdued by h = x, e = x, e = x, e ( i, j =, ; h = h ). () uiu j ui u j u j ui Then for a fixed oint = x ( u, u) of S, the signed distane from a nearby oint 5

6 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Differential Geometry - Taashi Saai u du u du x ( +, + ) to the tangent lane i, j= i j TS is ontrolled by half of II = h ( u, u ) du du. II is alled the seond fundamental form of S, and is reserved under orientation reserving arameter transformations of S. If II is definite at, i.e. its disriminant D h h h where is alled an elliti oint. If = 4( ) < 0, S lies on a one side of sides of TS where is alled a hyerboli oint. TS around D= 4( h h h ) > 0 at, S is loated on both Now to measure how S is urved in around a oint S, tae urves : s x() s = x( u() s, u()) s on S arameterized by ar length with x (0) =. For suh a urve, the normal omonent of x (0), alled the normal urvature, is given by κ = x(0), e = h ξξ with x (0) = ξ x, g ξξ =. n i, j= i j i= i ui i, j= i j One onsiders the minimum and the maximum of normal urvatures h ξ ξ i, j i j at under the ondition g ξξ i j = that are alled the rinial urvatures of S at. i, j Now the mean urvature H( = H S ) and the Gaussian urvature K( = K S ) of S at are defined as the arithmeti mean and the rodut of rinial urvatures, resetively. They are given by K = det( h ) det( g ), H = g h + g h g h g = g h ( ) {det( )} tr(( ) ( )) () where tr( a ) = a + a is the trae. Gaussian urvature (res. mean urvature) is invariant under (res. orientation reserving) arameter transformations of S. For a lane one gets K = H = 0, and for a shere of radius r, K = / r, H = / r hold. Note that if K( ) > 0 (res. < 0 ), is an elliti (res. hyerboli) oint. There are variety of flat (i.e. K 0 ) surfaes. Flat surfaes (with H 0 ) obtained as a family of lines (ruled surfaes) are alled develoable surfaes inluding ones, irular ylinders, and tangent surfaes (onsisting of tangent lines to a sae urve). A surfae of revolution given by rotating a urve x = f( u), x = g( u) in xx -lane around x -axis is a tyial surfae for whih KH, tae simle forms. Figure. Gaussian urvature ( K > 0, K < 0, K 0 ) 6

7 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Differential Geometry - Taashi Saai Now a surfae S with H 0 is alled a minimal surfae, sine suh an S is stable with reset to the area. Belgian hysiist J. Plateau verified exerimentally that soa film obtained by diing a wire form in soa solution has suh a roerty. A surfae of revolution given by x = osh x is a tyial examle of minimal surfae alled the atenoid. Minimal surfaes have been extensively studied in relation to artial differential equations and funtion theory of a omlex variable: For examle, if f : is a C -funtion whose grah is a minimal surfae, then f is a linear funtion (Bernstein theorem); minimal surfaes are reresented by holomorhi and meromorhi funtions defined on unit oen dis D (Weierstrass reresentation formula). Figure. Minimal surfaes (atenoid, Enneer surfae, right helioid) Intrinsi geometry of surfaes It is ossible to exress the Gaussian urvature K in terms of the first fundamental quantities g and their artial derivatives u to the seond order, i.e. K is an intrinsi geometri invariant of S, as was shown and alled Theorema Egregium" by Gauss. The length of a urve on S is given by () in terms of g, and then the intrinsi distane between two oints on S is defined as the infimum of the lengths of urves on S joining them. Gauss develoed his surfae theory based on the distane where the Gaussian urvature lays an imortant role. Now, a urve γ : x() t = x ( u() t, u()) t on S suh that the orthogonal rojetion of the aeleration vetor x () t to Tx () t S vanishes everywhere is alled a geodesi. Then γ roeeds straight on S sine it annot feel any aeleration fore on S, and satisfies i i j, = j j u () t + Γ u () t u () t = 0 ( i = ), where l ( il, j jl, i, l ) ( ) ( ) and Γ = g g + g g / ; g = g l, " denotes the differentiation with reset to u ( =, ). A geodesis γ exists at least for small t, one the initial oint and diretion are given, where γ () t is onstant. Now for a geodesi triangle Δ (a simly onneted domain in S bounded by a losed urve onsisting of three geodesi segments), ABC,, denote the (inner) angles of Δ. Then one obtains the Gauss-Bonnet formula: 7

8 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Differential Geometry - Taashi Saai A+ B+ C π = KdS (4) Δ If K 0 this is a familiar theorem that the sum of angles of a triangle in Eulidean lane is equal to π, and if K > 0 (res. K < 0 ) on Δ the sum of angles of a triangle is greater (res. less) than π. Indeed a surfae S with K gives a loal model for hyerboli geometry, although the whole hyerboli lane annot be realized as a surfae in. In the above one onsidered a iee of a surfae, and now wants to define the whole surfae. A subset S of is alled a regular surfae of lass C if the following holds: For eah oint S there exist an oen neighborhood U of in and a arameterized C -surfae x : D that is a homeomorhism from D onto U S with the relative toology. Thus eah oint of S admits a oordinate system (or hart) given by a homeomorhism x : U S D, and the whole S is desribed by an atlas onsisting of suh harts. For two harts ( Ui S,x i ) ( i =, ) reresenting a oint of S, the oordinate transformation is given by x x: x ( U U) x ( U U) that is a C -ma between domains of lane. A omat regular surfae is alled a losed regular surfae (e.g. shere, torus T that is the surfae of a doughnut, and the surfae Σ of a doughnut with holes). Dividing a losed regular surfae S into finitely many geodesi triangles and alying the Gauss-Bonnet formula, one obtains the Gauss-Bonnet theorem reresenting a toologial invariant χ ( S), the Euler harateristi of S, in terms of Gaussian urvature (note that χ( Σ ) = ( ) ): χ( S) = KdA. π (5) S Here are some global results that haraterize the shere: If the Gaussian urvature K of a losed surfae is equal to a onstant r, then r > 0 and S is a shere of radius / r. If the mean urvature K of a losed surfae is equal to a onstant h, then h 0 and S is a shere of radius h. A losed regular surfae S in admits an elliti oint. If K > 0 everywhere, S is homeomorhi to shere and alled an ovaloid, sine S lies on the one side of TS at eah S and bounds a onvex set of.. Differentiable Manifolds Manifolds The notion of manifold was introdued by B. Riemann in 854 to set a foundation of geometry, i.e. how to gras the onet of sae that loally loos Eulidean sae, but may sread manifold in urved manner and be of higher dimension. In the language of modern mathematis, manifold is defined as follows: Let M be a Hausdorff toologial n sae. A air ( U, φ) of an oen set U of M and a ma φ : U is alled a hart with oordinate neighborhood U, if φ is a homeomorhism onto an oen subset of n Eulidean sae. A family of harts A = {( U α,φ α )} α A is alled an atlas of M if 8

9 MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Differential Geometry - Taashi Saai xi n i Bibliograhy Berger, M (00). A Panorami View of Riemannian Geometry xxiii Sringer-Verlag, Berlin, Heidelberg, New Yor, ISBN [This is a very readable boo telling about roblems and ideas of Riemannian geometry over 700 ages without giving detailed roof but with many itures and bibliograhies.] Gray, A. (998). Modern Differential Geometry of Curves and Surfaes with Mathematia, seond ed. xxiv CRC Press, Boa Raton, FL, ISBN [This is a boo integrating traditional differential geometry of urves and surfaes with new omuter demonstration aabilities to draw itures.] Gromov, M (999). Metri Strutures for Riemannian and Non-Riemannian Saes, x , Birhäuser Boston In., Boston, ISBN [In this remarable boo the author gives a universal viewoint of athing geometry as a whole on the basis of notions of metri and measure.] Kobayashi, S. and Nomizu, K. (996). Foundations of Differential Geometry, Vol.I, Vol.II (Rerints of the 96 and 969 originals, A Wiley-Intersiene Publiation) ix+9, x+468. John Wiley Sons In., New Yor, ISBN , [This is one of the advaned omrehensive textboos on differential geometry suitable for further study.] Shoen, R. and Yau S.-T. (994). Letures on Differential Geometry v International Press Inororated, ISBN [This is an advaned textboo on nonlinear analysis on manifolds and its aliation to differential geometry suitable for further study.] Biograhial Seth Taashi SAKAI born 94 in Toyo, Jaan, Professor Emeritus, Oayama University (Aril, 005 to date) Eduation: BS in Mathematis, Tohou University, Jaan (Marh, 96). MS in Mathematis, Tohou University, Jaan (Marh, 969). Ph.D. in Mathematis, Tohou University, Jaan (February, 97). Positions held: Instrutor, College of General Eduation, Tohou University, Jaan (Aril, 969-Marh, 97) Assistant Professor, College of General Eduation, Tohou University, Jaan (Aril, 97-Marh, 974) Assistant Professor, Deartment of Alied Mathematis, Faulty of Engineering, Kyushu University, Jaan (Aril, 974-Marh, 976) Assistant Professor, Deartment of Mathematis, Hoaido University, Jaan (Aril, 976-Marh, 98) Professor, Deartment of Mathematis, Oayama University, Jaan, (Aril, 98-Marh, 005) Professor, Deartment of Alied Mathematis, Oayama University of Siene, Jaan, (Aril, 005 to date) Gastrofessor (Visiting Professor), SFB, University at Bonn, Germany, (Otober, 977-Marh, 979) 48