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Differential Geometry - Takashi Sakai MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Differential Geometry - Takashi Sakai DIFFERENTIAL GEOMETRY Takashi Sakai Department of Applied Mathematics, Okayama University of Science, Japan Keywords: curve, curvature, torsion, surface, Gaussian curvature, mean curvature, geodesic, minimal surface, Gauss-Bonnet theorem, manifold, tangent space, vector field, vector bundle, connection, tensor field, differential form, Riemannian metric, curvature tensor, Laplacian, harmonic map Contents 1. Curves in Euclidean Plane and Euclidean Space 2. Surfaces in Euclidean Space 3. Differentiable Manifolds 4. Tensor Fields and Differential Forms 5. Riemannian Manifolds 6. Geometric Structures on Manifolds 7. Variational Methods and PDE Glossary Bibliography Bibliography Sketch Summary After the introduction of coordinates, it became possible to treat figures in plane and space by analytical methods, and calculus has been the main means for the study of curved figures. For example, one attaches the tangent line to a curve at each point. One sees how tangent lines change with points of the curve and gets an invariant called the curvature. C. F. Gauss, with whom differential geometry really began, systematically studied intrinsic geometry of surfaces in Euclidean space. Surface theory of Gauss with the discovery of non-Euclidean geometry motivated B. Riemann to introduce the concept of manifold that opened a huge world of diverse geometries. Current differential geometry mainly deals with the various geometric structures on manifolds and their relation to topological and differential structures of manifolds. Results in linear algebra (Matrices, Vectors, Determinants and Linear Algebra) and Euclidean geometry (Basic Notions of Geometry and Euclidean Geometry) are assumed to be known as aids in enhancing the understanding this chapter. 1. Curves in Euclidean Plane and Euclidean Space Plane curves 2 A curve c in Euclidean plane with orthogonal coordinates ()x12, x is regarded as the locus of a moving point with time and given by a parametric representation x()txtxtatb=,,≤≤. (12 () ()) (1) 321 ©Encyclopedia of Life Support Systems (EOLSS) MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Differential Geometry - Takashi Sakai k It is assumed that functions x12()txt, () are of class C (2)k ≥ and the tangent vector x()ttt=, (xx12 () ()) to c is nonzero at each t . Then the tangent line to c at x()t0 is given by tttt xx()00+ () in vector notation. The arc length st() and the length Lc() of c are defined as tt b 22 st()==+≤≤,=.xx () tdtxx12 () t () tdta ( t b ) Lc ( ) () tdt (2) ∫∫aa ∫ a Indeed, the length Lc() doesn’t depend on parameterizations of a curve. Since sst= () is a strictly increasing function, one may take the inverse function tts= () and gets a new parametric representation ssts→=xx() (()) of c by arc length s for which x()s ≡ 1 holds. Setting ex1()ss= () and e2 ()sss= (−,xx21 () ()), a unit normal vector to c given by rotating e1()s through 90 counterclockwise, one obtains a (positive) orthonormal basis {()ee12ss, ()} called the Frenet frame of c at each x(s ) . Curvature Let cssL:=xx()0 ,≤≤ be parameterized by arc length. Then the acceleration vector x()sss=, (xx12 () ()) of c is orthogonal to x()s , and one may write xe()ssss==e1 ()κ ()2 (), where κκ()(ss==,=c ())xe () ssssss2 ()xx12 () () − xx 21 () () is called the curvature of c at x(s ) . Setting ρ(ss )= 1/κ ( ) if κ (s )≠ 0 , the circle centered at xe()sss+ ρ ()2 () (center of curvature) of radius ρ()s is tangent to c of second order at x()s . The centers of curvature of c form a curve called the evolute of c . For a curve ct:=xx() parameterized by t , one obtains 2232 κ()ttttttt=−{}xx12 () () xx 12 () (){ x 1 () +. x 2 ()} (3) Here is an example: The locus of a fixed point on the circle of radius a rolling on x1 - axis is called the cycloid and given by xt12()= at (−, sin) t xt () =−,≤≤ a (1cos)0 t t 2π . Its arc length and curvature are respectivelyst ( )= 2 a {1−/ cos( t 2)} and κ ()tat=− 1{4sin(2)} / / . The evolute of a cycloid c defined for −∞<t <∞ is again a cycloid that is congruent to c . What is the meaning of curvature? It appears in the Frenet formula: xe()ssssssss=,12 ()ee12 () =κ () e () , () =−κ () e 1 () (4) that controls the local behavior of the Frenet frame and the curve itself. Now an angle θ ()s ∈R between the unit tangent vector x(s ) to c and a fixed unit vector may be defined so that ss→θ ( ) is of class C k−1 and one has x(ss )=+,+ (cos(θ ( )αθα ) sin( ( s ) )) . Then κ (ss )=θ ( ) holds, i.e. curvature is an intrinsic invariant of a curve. Indeed, curvature determines the curve: Suppose a C k−2 - 322 ©Encyclopedia of Life Support Systems (EOLSS) MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Differential Geometry - Takashi Sakai function κ (s ) is given on [0, L ] . Then there exists a unique C k -curve cs:=xx( ) in 2 parameterized by arc length with κ c (ss )≡ κ ( ) , up to parallel translations and rotations of 2 . In the standard case of κ (s )≡ 0 (resp. κ (sk )≡ ≠ 0), c is a (part of a) straight line (resp. circle of radius 1 k ). A curve given by the equation κ (ssa ) =/ (a ; a nonzero constant) is called a clothoid that describes the trajectory of a car running with unit speed and an increasing acceleration of the constant rate, and applied to the design of highways. Figure 1. Cycloid and its evolute (left); clothoid (right) Let cL:,[0 ] →2 be a closed curve, i.e. xx()kk(0)= () (L ) hold for k -th derivatives L ( 02≤≤k ). The integral κ ()sds is called the total curvature of c , and the rotation ∫0 L number of c is given by (1/ 2πκ ) (sds ) that turned out an integer. For a simple (i.e. ∫0 without self-intersection points) closed curve c , the rotation number is equal to ±1, and two closed curves are deformed to each other (by regular homotopy) if and only if they have the same rotation number. A simple closed curve c admits at least four vertices at which the derivative κ()s vanishes. Space curves 3 A curve c in Euclidean space =,,∈{( xxx123) xi } is given by x()t=,,,≤≤ ( xtxtxt123 () () ()) a t b (5) using a parametric representation. It is assumed that xi (t ) (1≤ i ≤ 3) are of class k C ( k ≥ 3) and the acceleration vector x()tttt= (xxx123 (),, () ()) is linearly independent of the tangent vector x()tttt=,, (xxx123 () () ()). Arc length sst= () is defined by (2) and one gets the parameterization of c by arc length. Then ex1()ss= () is a unit vector and x()s is orthogonal to x(s ) . Now the curvature κ ()(ss= κ c ()) of a space curve c is 323 ©Encyclopedia of Life Support Systems (EOLSS) MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Differential Geometry - Takashi Sakai defined as x()s that is assumed to be positive. Set ex2 ()sss= ()/κ () and consider the vector producteee312()sss=× () (). One obtains a (positive) orthonormal basis {()}eiis 13≤≤ called the Frenet frame at each point x(s ) of the curve. Then the following Frenet-Serret formula ⎧e1()sss= κ ()e () ⎪ 2 ⎨e 2()sssss=−κτ ()ee13 () + () () (6) ⎪ ⎩e 3()sss=−τ ()e2 () 2 holds, where ττ()(ss==,=c ())e 2 () ssexxxx3 () det(() ssss ,, () ()) () is called the torsion of c (det means the determinant of 33× -matrix formed by the components of three vectors). The curvature and the torsion of a curve xx= (t ) parameterized by t are given by 32 κτ()tttt=×xx () () x () , () t = det(()xxxxx ttttt ,, () ()) () × () . (7) Once κτ()sssL>, 0 ()(0 ≤ ≤ ) are given, there exists a unique curve c parameterized by arc length with κ cc(ssss )=,κτ ( ) ( ) = τ ( ) , up to parallel translations and rotations of 3 . c is a plane curve if and only if its torsion vanishes everywhere, and a curve with constant curvature κ (sa )≡> 0 and constant torsion τ (sb )≡ ≠ 0 is congruent to a regular helix given by x(s )=+ ( ab221 )− ( a cos( absa 22 +, ) sin( absbabs 22 +, ) 22 +. ) L For the total curvature κκc = ()sds of a closed curve cssL:=xx( )(0 ≤≤ ) , κπc ≥ 2 ∫0 holds . Moreover if c is a knot, κπc > 4 holds . 2. Surfaces in Euclidean Space Fundamental forms and curvature In nature one sees various curved surfaces and nowadays curved surfaces are put use to designs, e.g. for cars. In differential geometry, one treats parameterized surface S given by a map 3 xx:∋D()()(()()()) uu12 , uu 12 , = xuu 112 , , xuu 212 , , xuu 312 , ∈ (8) 3 from a domain D of uu12-plane into , where xi ()uu12, (1≤ i ≤ 3) are functions of k class C ( k ≥ 2 ). For example, the graph of a function x312= fxx(), is expressed as 2222 ()(x12,→,,,xxxfxx 12 ()) 12. The sphere x123+ xxr+= of radius r is given by xr11221231=,=,=;−/<</,≤≤.cos u cos uxr cos u sin uxr sin uπ 2 u 1ππ 2 0 u 2 2 To 324 ©Encyclopedia of Life Support Systems (EOLSS) MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Differential Geometry - Takashi Sakai guarantee that S is a 2-dimensional figure, one takes curves uuu112 x(), , uuu212 x(), on S fixing uu21, respectively and assumes that the tangent vectors to these parameter curves ⎛⎞∂∂xx∂x x uu, =12 uu ,, uu ,,3 uu , i =, ui ()12⎜⎟ ()()(),(12) 12 12 12 ⎝⎠∂∂∂uuuiii are linearly independent and span the tangent plane TSp to S at each p =,x()uu12. Namely, the vector product xx()()0uu, ×,≠ uu everywhere, and one may take uu1212 12 ex()uu,= ()()()() uu ,× x uu , x uu ,× x uu , , (9) 12uu1212 12 12 uu 12 12 a unit normal vector to S .
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