Chapters 2 through 7 contain the core mathematical content. The text follows the Erlangen Program, which develops **geometry** in terms of a space and a group of transformations of that space. Chapter 2 focuses on the complex plane, the space on which we build two-dimensional **geometry**. Chapter 3 details transformations of the plane, including Möbius transformations. This chapter marks the heart of the text, and the inversions in Section 3.2 mark the heart of the chapter. All non-Euclidean transformations in the text are built from inversions. We formally dene **geometry** in Chapter 4, and pursue hyperbolic and elliptc **geometry** in Chapters 5 and 6, respectively. Chapter 7 begins by extending these geometries to dierent curvature scales. Section 7.4 presents a unied family of geometries on all curvature scales, emphasizing key results common to them all. Section 7.5 provides an informal development of the **topology** of surfaces, and Section 7.6 relates the **topology** of surfaces to **geometry**, culminating with the Gauss-Bonnet formula. Section 7.7 discusses quotient spaces, and presents an important tool of **cosmic** **topology**, the Dirichlet domain.

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Abstract: From a mathematical perspective, a surface is a generalization of a plane which does not necessarily require being flat, that is, the curvature is not necessarily zero. Often, a surface is defined by equations that are satisfied by some coordinates of its points. A surface may also be defined as the image, in some space of dimensions at least three, of a continuous function of two variables (some further conditions are required to insure that the image is not a curve). In this case, one says that one has a parametric surface, which is parametrized by these two variables, called parameters. Parametric equations of surfaces are often irregular at some points. This is formalized by the concept of manifold: in the context of manifolds, typically in **topology** and differential **geometry**, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane. A parametric surface is the image of an open subset of the Euclidean plane by a continuous function, in a topological space, generally a Euclidean space of dimension at least three. The paper aims at giving an **introduction** to the theory of surfaces from differential **geometry** perspective.

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We shall be using John’s Theorem several times in the remaining lectures. At this point it is worth mentioning important extensions of the result. We can view John’s Theorem as a description of those linear maps from Euclidean space to a normed space (whose unit ball is K) that have largest determinant, subject to the condition that they have norm at most 1: that is, that they map the Euclidean ball into K. There are many other norms that can be put on the space of linear maps. John’s Theorem is the starting point for a general theory that builds ellipsoids related to convex bodies by maximising determinants subject to other constraints on linear maps. This theory played a crucial role in the development of convex **geometry** over the last 15 years. This development is described in detail in [Tomczak-Jaegermann 1988].

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An Axiomatic system is a set of axioms from which some or all axioms can be used in conjunction to logically derive a system of **Geometry**. In an axiomatic system, all the axioms that are deﬁned must be consistent where there are no contractions within the set of axioms. The ﬁrst mathematician to design an axiomatic system was Euclid of Alexandria. Euclid of Alexandria was born around 325 BC. Most believe that he was a student of Plato. Euclid introduced the idea of an axiomatic **geometry** when he presented his 13 chapter book titled The Elements of **Geometry**. The Elements he introduced were simply fundamental geometric principles called axioms and postulates. The most notable are Euclid ﬁve postulates which are stated in the next passage.

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The development in the previous section concentrated on expected **geometry**. As we have seen this is not the only sensible or important geometrical structure for a parametric family of distributions. However the structure which was developed does seem to have a general ap- plicability across most of the possible geometric structures. This was recognised by Lauritzen (1987). He defined a general structure which encompassed most forms of statistical **geometry**. A statistical manifold is defined as (M, g, T ) where M is a manifold of distribution functions, g is a metric tensor and T is a covariant 3-tensor which is symmetric in its components and called the skewness tensor. For this structure there will always be set of a"ne connections which parallels the structure in the previous section.

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We will start with the example of the planar manipulator with three revolute joints. The manipulator is called a planar 3 R manipulator. While there may not be any three degree of freedom (d.o.f.) industrial robots with this **geometry**, the planar 3 R **geometry** can be found in many robot manipulators. For example, the shoulder swivel, elbow extension, and pitch of the Cincinnati Milacron T3 robot (Figure 2) can be described as a planar 3 R chain. Similarly, in a four d.o.f. SCARA manipulator (Figure 8), if we ignore the prismatic joint for lowering or raising the gripper, the other three joints form a planar 3 R chain. Thus, it is instructive to study the planar 3 R manipulator as an example.

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Completion of dimension n regular local ring with this property (that the residue field is contained in ring) is isomorphic to k[[t 1 ,.. , tn]].[r]

Abstract: The Cartesian word or “Cartesianity” was born with the philosophy of Descart (1596 - 1650). He was at the base of a doctrine based on rationalism, that it is means the search for truth by reason. Among others, Sigmend Freud had also approached this notion of psychological point to study the enigma of thoughts in humans. Other aspects of the Cartesian word have been used in mathematical **geometry**, namely cartesian coordinates and Cartesian referentials. As you know, studying a shape with curved and enclosed borders is more complicated than working on shapes with linear borders without curvature. In the way, we will introduce to the Cartesian **geometry** and characterize he Cartesian shapes.

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There are many texts on differential geome- try, many of which use diagrams to illustrate the concepts they are trying to portray, for example [1–8]. They probably date back to Schouten [8]. The novel approach adopted here is to present as much differential geom- etry as possible solely by using pictures. As such there are almost no equations in this ar- ticle. Therefore this document may be used by first year undergraduates, or even keen school students to gain some intuition of dif- ferential **geometry**. Students formally study- ing differential **geometry** may use this text in conjunction with a lecture course or standard text book.

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In this article, we focus on the multi-view images context. In order to obtain superpixels that are co- herent with the scene **geometry**, we propose to inte- grate a geometric criteria in superpixels construction. The proposed algorithm follows the same steps as the well known SLIC, Simple Linear Iterative Clustering approach (Achanta et al., 2012) but the aggregation step takes into account the surface orientations and the similarity between two consecutive images. In §2, we present a brief state of the art on superpixels construc- tors. Then, an overview of the proposed framework is presented, followed by the extraction of geometric in- formation and its integration in a k-means superpixels constructor. Finally, experiments on synthetic data are presented.

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As we have seen in the last chapter, the cdcd is a very powerful tool. But it does not give suﬃcient control on the relative disposition of the cells, when they are not contained in the same cylinder. In particular, one cannot, in general, reconstruct the **topology** of a deﬁnable set from its decomposition into cells of an adapted cdcd. The main diﬃculty is that we have no control on how a deﬁnable continuous function ζ : C → R on a cell C behaves as one approaches the boundary of C. The function ζ, even if it is bounded, may not extend to a continuous function on clos(C). For instance, the deﬁnable continuous function ζ deﬁned on the set of (x, y) ∈ R 2 such that x > 0 by ζ(x, y) = 2xy/(x 2 + y 2 ) does not extend continuously to (0, 0). All points (0, 0, z) with − 1 ≤ z ≤ 1 belong to the closure Γ of the graph of ζ. The deﬁnable set Γ ⊂ R 3 has

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Fractal **geometry** is based on the idea of self-similar forms. To be self- similar, a shape must be able to be divided into parts that are smaller copies which are more or less similar to the whole. Because of the smaller similar divisions of fractals, they appear similar at all magnifications. However, while all fractals are self-similar, not all self-similar forms are fractals. (For example, a straight Euclidean line and a tessellation are self-similar, but are not fractals because they do not appear similar at all magnifications). Many times, fractals are defined by recursive formulas. Fractals often have a finite boundary that determines the area that it can take up, but the perimeter of the fractal continuously grows and is infinite.

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Remark 3.4 One can define N-graded manifolds as Z-graded manifolds whose dimen- sion is indexed by N. Therefore, any homogeneous section has non-negative degree. This leads to an interesting property: on an N-graded manifold, there does not exist a section of degree zero which can be obtained as a product of sections of non-zero degree. This means that the locality and the π -locality of the stalks are equivalent con- ditions. Hence, N -graded manifolds do not require the **introduction** of formal series to be studied, unlike Z-graded manifolds (see Remarks 2.19 and 3.2). Another difference between N- and Z-graded manifolds is that there exists a Batchelor-type theorem on N-graded manifolds [3]. It is not known if an analogous result holds for Z-graded manifolds.

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Since the work of Donaldson in the 1980s, gauge theory has evolved into an indispensable tool in the study of smooth 4-manifolds. After the **introduction** of the Seiberg- Witten invariants in 1994, the theory was both simplified and extended, and the relation to Gromov-Witten in- variants (for symplectic 4-manifolds) was established by Taubes. The equivalence of the Donaldson and Seiberg-Witten invariants, conjectured by Witten in 1994, now seems close to being proved by Feehan-Leness. Kronheimer-Mrowka have used part of the work of Feehan-Leness combined with that of Taubes to complete the proof of Property P for knots; later they reproved this more directly using sutured Floer homology. Recently, the Seiberg-Witten invariants together with the rational blow- down technique of Fintushel-Stern have been applied to prove the existence of exotic smooth structures on CP²#k (-CP²) for small values of k according to work of Akhmedov-Park after initial results of J. Park.

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tinct [98], proposals also emerged for gapping graphene and realizing assorted topo- logical effects. Because the spin-orbit coupling in graphene is quite weak, experi- mental proposals to gap it have relied on patterning substrates or applying external fields. In single-layer graphene, applying a boron-nitride substrate provides an on- site potential that gaps the Dirac points without coupling the valleys (at least to the extent that the two lattices are made perfectly commensurate). In this construction, the different signs of the mass term, corresponding to placing a boron atom on the A or the B sublattice of graphene, correspond to different halves of the Brillouin zone having different Chern numbers [103, 123]. This difference in valley Chern number implies that a domain wall between two oppositely gapped heterostruc- tures should host domain wall states [1, 103]. However, unlike the Chern numbers in the quantum Hall effect discussed in the beginning of this **introduction**, these Chern numbers are not strictly topological. If the system Hamiltonian is deformed in such a way that the valleys are coupled, then the half-Brillouin-zone Chern num- ber is no longer quantized, and topological edge states are no longer guaranteed. In this sense, I personally consider any discussion of **topology** in graphene or graphene bilayers “quasi-topological.” While the tools and language of topological condensed matter physics can be used to describe some properties of graphene, one should not get too carried away in proposing a new transport signature that in real systems gets washed out by small, but highly relevant, valley nonconserving disorder.

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The consistent patterns of galactic rotation curves using precise and independent galactic redshift data confirmed that the hydrogen clouds and outer stars are orbiting galaxies at speeds faster than that calculated using Newtonian laws. Accordingly, the dark matter hypothesis was introduced to account for the apparently missing galactic mass and to explain the fast-orbital velocity [32, 33]. However, no evidence of the existence of the dark matter, which is supposed to account for the majority galactic mass, was observed since its **introduction**. The failure to find dark matter led to the **introduction** of new theories such as modified gravity and modified Newtonian dynamics [21, 22, 34 - 36]. On the other hand, several recent studies found that many galaxies do not contain dark matter [37 – 39]. This observation was considered in some studies where the galaxy formation was simulated using modified Newtonian dynamics without considering the dark matter [40]. Thus, it seems that there is no evidence or agreement on the existence or nature of the dark matter as well as it is not an essential element in some galaxies. As an alternative, I introduce a new hypothesis based on the variation of the curvature of both sides along with the evolution of the horizon radius as was shown Figure 4, where the curvature of the spacetime varies at the different instant of the time. Accordingly, I

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The consistent patterns of galactic rotation curves using precise and independent galactic redshift data confirmed that the hydrogen clouds and outer stars are orbiting galaxies at speeds faster than that calculated using Newtonian laws. Accordingly, the dark matter hypothesis was introduced to account for the apparently missing galactic mass and to explain the fast-orbital velocity [32, 33]. However, no evidence of the existence of the dark matter, which is supposed to account for the majority galactic mass, was observed since its **introduction**. The failure to find dark matter led to the **introduction** of new theories such as modified gravity and modified Newtonian dynamics [21, 22, 34 - 36]. On the other hand, several recent studies found that many galaxies do not contain dark matter [37 – 39]. This observation was considered in some studies where the galaxy formation was simulated using modified Newtonian dynamics without considering the dark matter [40]. Thus, it seems that there is no evidence or agreement

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Symplectic structures and contact structures are closely related. They share the same property of having no local invariants (Darboux Theorem). In some special cases, sym- plectic structures and contact structures can be translated from one to another by the processes of symplectification and contactification. In recent years, exploration of this transition is proved to be one of the most fruitful approaches in the study of contact **geometry**.

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We observe that many tasks in sensor networks may alternatively be stated in topo- logical terms. The tasks of detection and localization of coverage holes and worm holes are two such examples. The combinatorial information mentioned above is suﬃcient to compute topological invariants, and is the subject of Algebraic **Topology**. We give a brief **introduction** to Algebraic **Topology** in Chapter 2 and employ this theory to develop dis- tributed algorithms to detect and localize coverage and worm holes. We emphasize that this is the ﬁrst work which simultaneously solves both problems, supporting our thesis that algebraic **topology** oﬀers a general framework for topological analysis in sensor net- works.

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Thus one of the main and fundamental areas of geo- metry, projective **geometry**, was developed, first by the French (Carnot 1803) and then by the English, Germans and Italians in the nineteenth century. The Frenchman Poncelet wrote notes on **geometry** in a Russian military prison from 1812–1814, followed by the Germans Möbius (1828), Grassmann the elder (1844) and Karl Von Staudt (1847), the nobleman from Rothenburg ob der Tauber (see Plaumann & Strambach (1981) for Staudt’s innovative point of view). Following this tradition came Felix Klein and his “Erlangen Program” (Klein 2004). This program says that groups of symmetries are equivalent to the various kinds of **geometry**. It had all kinds of developments, including the use of group representations in physics to represent particles. In Klein (1926–27) there is also a valuable history. Most groups involved with physics are “Lie” or continuous groups; Sophus Lie was a Norwegian associate of Klein. In the twentieth century the **geometry** was generalised, with technical details such as linear dependence properties being better explained by Whitney’s theory of matroids. Late in the nineteenth century, prompted by the foundational work of the Englishman Cayley and the German Riemann (1854), Klein, Poincaré, Minkowski, then Einstein incorporated higher-dimensionality to explain space-time properties like relativity and gravity. The speed of light is basically constant, and space is deformed by mass, a property that can be explained by assuming the equivalence of spatial **INTRODUCTION**

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