(MTH5109) GEOMETRY II: KNOTS AND SURFACES LECTURE NOTES. 1. Introduction to Curves and Surfaces

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1 (MTH509) GEOMETRY II: KNOTS AND SURFACES LECTURE NOTES DR. ARICK SHAO. Introduction to Curves and Surfaces In this module, we are interested in studing the geometr of objects. According to our favourite source Wikipedia [9], geometr is the branch of mathematics concerned with studing the shapes, sizes, and positions of objects. Throughout this term, ou will develop a better understanding of these concepts, as well as eplore how these can be quantified and computed. In our mathematical education up to this point for instance, in calculus and linear algebra ou have considered flat, linear spaces such as the real line (R), the plane (R 2 ), and more general n-dimensional spaces (R n ). This module will epand our outlook to curved objects... Some Burning Questions. Before launching into more involved mathematical discussions, we first address some common questions that ou ma have regarding this module, its contents and features, and how it fits into our overall maths education. Question.. Wh would we want to stud curved objects? Curved objects are everwhere in our lives. If ou throw a ball or shoot a missile into the air, then the trajector of the ball or missile will not be linear, but rather a curve (due to gravit, for eample). The surface of the earth is not flat, but rather like a sphere. To stud this surface as a whole, we have to understand the effects of its curvature. This is important for man questions, such as determining the shortest flight path between two cities. According to Einstein s landmark theor of general relativit, the universe that we inhabit is not flat, but rather a 4-dimensional curved object ( spacetime ). Moreover, gravit itself is modelled b the shape and curvature of the spacetime. These are onl a small sample of motivations for having a firmer understanding of geometr. However, since this is an elementar module, and since we have limited time, we will onl discuss: -dimensional objects: curves. 2-dimensional objects: surfaces. 4-dimensional spacetimes and gravit will have to wait until another module; if ou are interested in such things, ou should consider the third-ear module MTH632: Relativit. Question.2. What maths will we use in this module? In other words, what should I know? This module is mainl concerned with the differential geometr of curves and surfaces. In particular, we look at objects that are without jagged edges and var smoothl enough so that

2 2 (MTH509) LECTURE NOTES one can take derivatives, or linear approimations. In addition, to measure the sizes e.g. length and area of objects, we will need need to compute various integrals. As a result, this module will assume that ou have moderate familiarit with first-ear calculus, for which differentiation and integration are two cornerstones. When studing -dimensional curves, we will make frequent use of single-variable calculus (mainl contents from MTH400/4200: Calculus I). When studing 2-dimensional surfaces, we will require some knowledge of partial derivatives and double integrals (both of which ou encountered in MTH40/420: Calculus II). The simplest objects we can consider are lines and planes; these fall under the stud of linear algebra. In particular, we will sometimes reference a bit of background on vectors and matrices: As we are working in one and two dimensions, the linear algebra background ou need should have been covered in MTH403/4203: Geometr I. Most of ou are also currentl learning this in more detail in MTH52: Linear Algebra I. Even for more comple curved objects, a useful tool in their analsis is linearisation first determining and studing the linear object that best approimates it. Thus, we cannot reall escape the need to understand linear algebra and its connections to geometr. Finall, in order to construct common eamples, we will make use of man elementar functions: Polnomials (e.g. t 2 + ), and rational functions. Trigonometric functions (sin, cos). Eponential functions (ep) and logarithmic functions (ln). Rarel, we ma encounter others, such as hperbolic functions (sinh, cosh). We will at times reference a few basic properties of these functions that ou have learned before. If ou want to be optimall prepared for the material ou will see in this module, it is recommended that ou revise the material in calculus and linear algebra ou have previousl learned. Question.3. Help! What will I be epected to learn? The main focus of the module is on the interface between some mathematics ou have alread encountered most notabl, calculus and linear algebra with concepts in geometr. For eample, ou will be epected to understand how notions such as derivatives, integrals, vectors, and matrices connect to the shapes and sizes of various objects ou will encounter. This part of the knowledge is largel conceptual in nature; it is more about understanding the material in a critical wa rather than memorising definitions, formulas, or algorithms. As the module is also concerned with quantifing geometric properties, ou will be epected to demonstrate that ou are capable with performing various tpes of computations. Again, these computations will involve elements of calculus (e.g. derivatives and integrals) and linear algebra (e.g. vector and matri algebra). In addition, ou will be epected to graph various curves and surfaces, either in a plane or in space. While our hand-drawn figures need not be equisite, ou will need to produce convincing and largel accurate depictions of these objects. Finall, though we will encounter a number of proofs in our discussions, the will not be a central focus of this module. In particular, ou will not be asked to memorise and recite length

3 (MTH509) LECTURE NOTES 3 proofs of various results that ou will encounter. On the other hand, ou will be required to have a basic understanding of wh these results are true..2. Some Informal Ideas. Here, we give a brief and informal introduction to the ideas we will eplore and the questions we will answer in this module. This is to get ou to begin engaging and thinking criticall about basic concepts. More detailed discussions will be given in later chapters..2.. Shape and Curvature. Consider the three curves in the plane in Figure. below. You probabl alread have a ver strong intuitive feeling that the curve C is straight, while and C 3 are curved. But, how might ou mathematicall describe this? Moreover, driving along a road, ou feel a difference between a sharp turn along a ver curved road and a smaller turn. Similarl, from the figure, ou probabl have a sense that C 3 is more curved than. But, how would ou capture and quantif this mathematicall? C C 3 z C 4 Figure.. C,, C 3 are curves on a plane, while C 4 is a curve in space. Another question concerns the direction of curvature. For instance, continuing with the driving analog, ou can distinguish between turning left and turning right. Similarl, for and C 3, depending on which direction ou traverse these curves, ou would have an intuitive feeling of whether the curve is bending anticlockwise or clockwise. Moreover, if the curve is situated in 3-dimensional space (for eample, see the heli C 4 in Figure.), then there is an additional dimension of directions that the curve could bend. Again, the question of interest is how we can describe and quantif all of this mathematicall. Moving on to surfaces, the situation is similar but even more complicated. Consider, for instance, the surfaces in Figure.2. Again, ou can distinguish that S is flat, while S 2, S 3, and S 4 are curved. You can also tell when a surface is ver curved, as opposed to slightl curved. What is novel for these 2-dimensional surfaces, in contrast to -dimensional curves, is that a surface can bend in different was along different directions. For instance, at an point of the spheres S 2 and S 3, the surfaces bend inward toward itself in the same wa no matter which direction ou go. However, for the saddle S 4, depending on which direction along the surface ou look, it could be bending in opposite directions. The mathematical challenge here, then, is to somehow capture and quantif all this geometric information.

4 4 (MTH509) LECTURE NOTES z S z S 2 z S 3 z S 4 Figure.2. S, S 2, S 3, S 4 are surfaces in 3-dimensional space Size, Length, and Area. Another aspect of geometr is studing the sizes of objects. Consider the two circles in Figure.3. Your intuition alread indicates that while the both have similar circular shape, the also have different sizes. You probabl also have enough mathematical background to know that ou can capture this b measuring their arc lengths (i.e., their circumferences). There is an analogous sense of size for surfaces; for instance, the spheres S 2 and S 3 in Figure.2 have similar shapes but different sizes. This can be similarl captured b measuring their surface areas. The mathematical goal, then, is to understand C Figure.3. C and are circles with different circumferences. how we define and compute these arc lengths and surface areas. From calculus, ou should know that lengths and areas are generall evaluated using integrals, and ou should also know how to integrate along a line segment or a (flat) region in a plane. The questions of arc length and surface area will now force us to make sense of what it means to integrate along a curve or along a surface Intrinsic and Etrinsic Geometr. While we will not be particularl precise here, we will also touch upon the following classification of geometric properties: Etrinsic properties are those that depend on how an object is situated in a larger space.

5 (MTH509) LECTURE NOTES 5 Intrinsic properties are of the object itself, regardless of how it is embedded in space. For eample, consider the unit circles in Figure.4. Clearl, these are different objects in terms of etrinsic geometric properties; after all, C and lie in a plane, while C 3 lies in 3-dimensional space. Moreover, while C and are in the same plane, the are situated at different locations. On the other hand, ou probabl have a sense that these three circles are somehow the same, regardless of what larger space the sit in or where in the space the sit. This intuition is behind the rough notion that the unit circles C,, C 3 have the same intrinsic geometr. 2 C 0.5 z C Figure.4. Three unit circles in different locations and settings. One thought eperiment ou can tr to further eplore intrinsic geometr is to imagine if ou are a bug living on one of these unit circles, with no knowledge of the larger dimensional space that the circle is in. As this imperceptive bug that knows onl of the circle itself, ou would not be able to distinguish between whether ou were on C,, or C 3. Similarl, two copies of the sphere S 2 in Figure.2 in different positions would be etrinsicall distinct, but also the same in a similar intrinsic sense. For a more compelling eample, consider the two curves in Figure.5. Clearl, the are etrinsicall different, since the are embedded ver differentl in the plane. On the other hand, imagine again that ou are the provincial bug. On either C or, ou have eactl two options: move in one direction, or move in the other. That is bending and C is not is purel a feature of how the are situated in the plane, and would not be evident to the bug that is unaware of the plane. As a result, an infinite curve is intrinsicall equivalent to a line! In contrast, consider a bug on a circle (e.g. C in Figure.3) and a bug on a line (e.g. C in Figure.5. A bug on C Figure.5. Two infinite curves that are etrinsicall different but intrinsicall the same.

6 6 (MTH509) LECTURE NOTES the circle that walked in one direction long enough would return to where it started, whereas a bug on a line would not. Thus, one can argue that a line and a circle are intrinsicall different. Similarl, consider two circle with different radii, for instance, C and in Figure.3. A bug would again be able to distinguish between them, since a bug on would need to travel less far to loop back to the starting point than a bug on C would. Eercise.. What about non-circular loops, such as that in Figure.6? What geometric properties of these loops are intrinsic? Can ou classif all such loops based on their intrinsic geometr? For surfaces, the full stor is far more comple, since the eistence of an etra dimension allows for man more intrinsic properties. We will stud some interesting ones later in the module. In contrast to curves, some aspects of how a surface is curved are in fact intrinsic! Unfortunatel, we will not have the time in this module to precisel define what we mean b intrinsic and etrinsic. (These notions are formall captured b mathematical objects known as manifolds.) However, this discussion is meant to encourage ou to think criticall along these lines and to keep these notions in mind as we discuss various geometric properties in detail. Figure.6. A non-circular loop Topolog. Consider now the sphere S 2 in Figure.2 and the egg S in Figure.7. Clearl, the sphere and the egg have different geometries (in fact, both etrinsicall and intrinsicall). On the other hand, there is an even weaker sense that these two surface are the same. One wa to think of this is that, if ou visualise the objects as elastic, then ou can stretch and compress the sphere in order to deform it into the egg. z S z S 2 Figure.7. An ellipsoid (S ) and a torus (S 2 ). Consider in addition the doughnut S 2 in Figure.7. In this same sense of deformations, the doughnut is distinct from the sphere and the egg. Indeed, ou cannot deform the sphere into the doughnut as ou did into the egg; for this, ou would have to be much more drastic b poking our finger through the sphere and puncturing a hole. However, ou can deform this doughnut

7 (MTH509) LECTURE NOTES 7 into the coffee mug on the left-hand side of Figure.8 (as clumsil indicated in that same figure). In this weaker sense, the doughnut and coffee mug are the same. Figure.8. Deforming a coffee cup into a doughnut! These intuitions are closel tied to an area of mathematics known as topolog. While we will not have the time to formall delve into topolog, we will in this module encounter some properties that are topological in nature. In other words, these properties do not depend on the shape or size of the object, but on other more basic features, such as how man holes a surface has. One particular application of topolog, in the one-dimensional setting, is to stud and classif knots. In what was can ou tie a rope into a knot? When are two knots the same? Where topolog enters is that intuitivel, what matters here is how the rope wraps around itself, and not the eact shape of this wrapping. Time permitting, we will discuss some elementar aspects of this knot theor near the end of the module.