Differential Geometry

In this paper we are concerned with the initial boundary value problems of linear and semilinear parabolic equations with mixed boundary conditions on non-cylindrical domains in spatialtemporal space. We obtain the existence of a weak solution to the problem. In the case of the linear equation the parts for every type of boundary condition are any open subsets of the boundary being nonempty the part for Dirichlet condition at any time. Due to this it is difficult to reduce the problem to one on a cylindrical domain by diffeomorphism of the domain. By a transformation of unknown function and penalty method we connect the problem to a monotone operator equation for functions defined on the non-cylindrical domain. In this way a semilinear problem is considered when the part of boundary for Dirichlet condition is cylindrical.

In this paper we construct a family of steady symmetric vortex patches for the incompressible Euler equations in a disk. The result is obtained by studying a variational problem in which the kinetic energy of the fluid is maximized subject to some appropriate constraints for the vorticity. Moreover, we show that these vortex patches shrink to a given minimum point of the corresponding Kirchhoff-Routh function as the vorticity strength parameter goes to infinity.

By variational methods, for a kind of Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature, we construct some multi-peak solutions as the parameter is sufficiently small under certain assumptions. We also obtain the asymptotic behaviors of the solutions.

By variational methods, for a kind of Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature, we construct some multi-peak solutions as the parameter is sufficiently small under certain assumptions. We also obtain the asymptotic behaviors of the solutions.

Newtonian gravity models gravitational interaction as a universal attractive force obeying an inverse square law effective across vast domains of engineering and astronomy while Einstein's general relativity reframes gravity as spacetime curvature, with freely falling bodies following geodesics rather than being acted on by a force a shift mandated by the equivalence of gravitational and inertial effects in local experiments. This work synthesizes both perspectives to separate apparent, acceleration induced "fake" gravity from curvature-based "real" gravity and to delineate their applicable regimes. Newtonian Gravity: Inverse-Square Interaction In Newton's framework, two masses attract with magnitude" 𝐹 = 𝐺" (𝑚_1 𝑚_2)/𝑟^2 , establishing a central, instantaneous, inverse-square force usable for orbital and near-Earth dynamics across moderate velocities and potentials. The inverse-square scaling arises from geometric dilution over spherical surfaces, and provides a compact, predictive model for trajectories, energy, and stability when relativistic corrections are negligible. Einstein's equivalence principle states that locally one cannot distinguish a uniform gravitational field from a uniformly accelerated frame operationally the physics in a small freely falling lab is identical to that of an inertial lab with no gravity and conversely standing at rest in a gravitational field mimics being in an accelerated rocket. This local indistinguishability motivates treating free fall as inertial motion in curved space time and identifies "weight" as a contact-force effect due to deviation from free fall not a direct gravitational pull on stationary bodies relative to the ground.

This paper clarifies the conceptual distinction between “real” gravity as spacetime curvature and “apparent” gravity as inertial effects due to acceleration, contrasting the Newtonian force picture with Einstein’s geometric framework. It formalizes the role of the equivalence principle, geodesic motion, and inverse-square behavior, and explains where and why Newtonian gravity remains an excellent approximation while general relativity becomes essential. Practical implications are framed around free fall, weightlessness, tidal effects, and long-distance trajectory deviations.

We rigorously prove that for compact charged general relativistic objects there is a lower bound for the mass-radius ratio. This result follows from the same Buchdahl type inequality for charged objects, which has been extensively used for the proof of the existence of an upper bound for the mass-radius ratio. The effect of the vacuum energy (a cosmological constant) on the minimum mass is also taken into account. Several bounds on the total charge, mass and the vacuum energy for compact charged objects are obtained from the study of the Ricci scalar invariants. The total energy (including the gravitational one) and the stability of the objects with minimum mass-radius ratio is also considered, leading to a representation of the mass and radius of the charged objects with minimum mass-radius ratio in terms of the charge and vacuum energy only.

The purpose of this paper is to try a formal consideration of gravity as a physical curvaturedeformation of a physical matter-free astro-region, generated by external to the astro-region field-energy attack. This attack-answer deformed but stable state of the astro-region suggests appropriate dynamical view on the physical structure of this region, based on existent real physical external disturbances, and on the real surviving answer-disturbances. Such a physical view on the situation: stress-answerstress, leads to correspondent attack-deformation formal theoretical view. Such physically deformed real physical region will demonstrate itself in two ways: physically and geometrically, i.e., new energy distribution, and new geometrical structure. Our purpose is these real deformation and surviving two aspects of the new physical stress-strain distribution to get formal tensor presentation. The spherically symmetric case will be considered from this physical-geometrical view.

The Source Energy Field Theory (SEFT) extends the freedom to define spacetime itself as a manifestation of the fundamental energy field. This enhanced degree of freedom allows the Standard Model gauge symmetries and the Higgs mechanism to emerge as natural geometric consequences, rather than imposed assumptions. While its application to cosmology has successfully explained supernova redshiftdistance relations, galactic rotation curves, and the cosmic microwave background, its implications for particle physics have remained unexplored. In this paper, we extend SEFT to the domain of particle physics and demonstrate that and preserving the custodial relation ρ = 1. These results establish SEFT as a framework in which the core architecture of the Standard Model is embedded in the dynamics of a single fundamental field, thus providing the particle-physics counterpart to the cosmological results previously derived in SEFT.

The complete set of concomitants is given of the metric, a scalar function, and their derivatives in six dimensions without imposing conditions on the order of the derivatives and their properties are studied under conformal transformations. These results are useful in several physical problems, like finding (9')"" and ( F")"' in any geometry.

We present a comprehensive framework for probabilistic modeling on Riemannian manifolds, encompassing diffusion processes, continuous normalizing flows, energy-based models, and information-theoretic measures adapted to curved geometries. Our unified approach extends classical probabilistic methods from Euclidean spaces to arbitrary Riemannian manifolds, providing principled tools for modeling data with inherent geometric structure. We develop complete mathematical foundations including forward and reverse stochastic differential equations, probability-flow ordinary differential equations, intrinsic Langevin dynamics, and manifold-aware information measures. The framework is demonstrated on canonical manifolds including spheres, rotation groups SO(3), symmetric positive definite matrices, and hyperbolic spaces, with applications spanning computer vision, robotics, neuroscience, and network analysis.

The aim of this paper is to obtain on the dual 1-jet space J 1 * (R, M ) the main geometrical objects used in the dual jet geometry of time-dependent Hamiltonians. We talk about distinguished (d-) tensors, time-dependent semisprays, nonlinear connections and their mathematical connections.

It is the purpose of the present paper to outline an introduction in theory of embeddings in the 2-osculator bundle. First, we recall the notion of 2-osculator bundle ([1], ) and the notion of submanifolds in the 2-osculator bundle. A moving frame is constructed. The induced connections and the relative covariant derivation are discussed in third and fourth sections.The Ricci identities for the deflection tensor are present in the next section.

In this paper, we develop the distinguished Riemannian differential geometry (in the sense of d-connections, d-torsions, d-curvatures, and the geometrical Maxwell-like and Einstein-like equations) for the time-dependent Hamiltonian of momenta which governs the electrodynamics phenomena.

summary:In this paper we develop the distinguished (d-) Riemannian differential geometry (in the sense of d-connections, d-torsions, d-curvatures and some geometrical Maxwell-like and Einstein-like equations) for the polymomentum Hamiltonian which governs the multi-time electrodynamics

The aim of this paper is to obtain on the dual 1-jet space J^{1*}(R;M) the main geometrical objects used in the dual jet geometry of time-dependent Hamiltonians. We talk about distinguished (d-) tensors, time-dependent semisprays, nonlinear connections and their mathematical connections.

By means of the spinorial representation of matrices, two constructed determinant-induced metrics of conformal-Euclidean Riemannian and of Finsler types, respectively, are shown to produce a (h, v)-structure, whose properties are investigated from the point of view of naturally developed Einstein and Maxwell-like equations. MSC2010: 54B40, 53C60.

In this article we present a study of the subspaces of the manifold OscM, the total space of the osculator bundle of a real manifold M. We obtain the induced connections of the canonical metrical N-linear connection determined by the homogeneous prolongation of a Finsler metric to the manifold OscM. We present the relation between the induced and intrinsic geometric objects of the associated osculator submanifold.

The aim of this paper is to create a large geometrical background on the dual 1-jet space J^{1*}(T,M) for a multi-time Hamiltonian approach of the electromagnetic and gravitational physical fields. Our geometric-physical construction is achieved starting only from a given quadratic Hamiltonian function of polymomenta H, which naturally produces a canonical nonlinear connection N, a canonical Cartan N-linear connection C\Gamma(N) and their corresponding local distinguished (d-) torsions and curvatures. In such a context, we construct some geometrical electromagnetic-like and gravitational-like field theories which are characterized by some natural geometrical Maxwell-like and Einstein-like equations. Some abstract and geometrical conservation laws for the multi-time Hamiltonian gravitational physical field are also given.

It is the purpose of the present paper to outline an introduction in theory of embeddings in the manifold Osc^{2}M. First, we recall the notion of 2-osculator bundle. The second section is dedicated to the notion of submanifold in the total space of the 2-osculator bundle, the manifold Osc^{2}M. A moving frame is constructed. The induced N-linear connections and the relative covariant derivatives are discussed in third and fourth sections. The Ricci identities of the Liouville d-vector fields are present in the last section.

In this paper we construct a distinguished Riemannian geometrization on the dual 1-jet space J^{1*}(T,M) for the multi-time quadratic Hamiltonian functions. Our geometrization includes a nonlinear connection N, a generalized Cartan canonical N-linear connection (together with its local d-torsions and d-curvatures), naturally provided by a given quadratic Hamiltonian function depending on polymomenta.

We prove that the Ricci scalar curvature and the Berwald scalar curvature of a two-dimensional Finsler space, considered over a vector field on the 3-dimensional flat space, are naturally related to 2-dimensional electro-capillary phenomena effects observed for a compressed monolayer. The Cartan tensor and the nonlinear Barthel connection of the Finsler model are determined, and the geometric objects which depend on compression speed and on the characteristics of the electrically charged double layer are used in order to reveal several classes of structure formation within the phase transition of the first order.

The authors study several distinguished geometrical properties of the time-dependent Lagrangians governing the 2D-motions of a particle of a monolayer. They apply several ideas of R. Miron and M. Anastasiei concerning Lagrangian geometry on the tangent bundle [The geometry of Lagrange spaces: theory and applications. Fundamental Theories of Physics. 59. Dordrecht: Kluwer Academic Publishers. (1994; Zbl 0831.53001)]. The considered curvature of the investigated slices is a Finsler flag curvature. The metric induced by a 2D monolayer Lagrangian is of Berwald-Lagrange curvature. It is proved that this is a scalar curvature, and its sign governs the geodesic deviation in the monolayer. They determine the components of the canonical semispray of the energy functional associated to the 2D monolayer Lagrangian, and the components of the Berwald N-linear connection. Also the physical meaning is explained.Reviewer: Lajos Tamássy (Debrecen)

We investigate thermodynamics for a magnetically charged regular black hole (MCRBH), which comes from the action of general relativity and nonlinear electromagnetics, comparing with the Reissner-Norström (RN) black hole in both four and two dimensions after dimensional reduction. We find that there is no thermodynamic difference between the regular and RN black holes for a fixed charge Q in both dimensions. This means that the condition for either singularity or regularity at the origin of coordinate does not affect the thermodynamics of black hole. Furthermore, we describe the near-horizon AdS 2 thermodynamics of the MCRBH with the connection of the Jackiw-Teitelboim theory. We also identify the near-horizon entropy as the statistical entropy by using the AdS 2 /CFT 1 correspondence.

Under the action of the c-map, special Kähler manifolds are mapped into a class of quaternion-Kähler spaces. We explicitly construct the corresponding Swann bundle or hyperkähler cone, and determine the hyperkähler potential in terms of the prepotential of the special Kähler geometry. Contents 1 Introduction 1 2 The c-map 2 3 Hyperkähler cones and the Legendre transform 3 4 Hyperkähler cones from the c-map 5 4.

This five booklet includes the corrections of 65 exercises and problems of the following nine chapters: topography, astronomy, curves and the theory of surfaces, ellipse and ellipsoïd, geodetic systems, reduction of distances, the map projections, the transformations of passage between the geodetic systems and notion of the movement of an Earth artificial satellite.

Some mutual relations between p-cyclic contractive self-mappings, p-cyclic Kannan self-mappings, and Meir-Keeler p-cyclic contractions are stated. On the other hand, related results about the existence of the best proximity points and existence and uniqueness of fixed points are also formulated.

This paper discusses a more general contractive condition for a class of extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are nonempty, closed and convex, then all the iterations converge to a unique closed limiting finite sequence, which contains the best proximity points of adjacent subsets, and reduce to a unique fixed point if all such subsets intersect.

In this paper, we introduce best proximal contractions in complete ordered non-Archimedean fuzzy metric space and obtain some proximal results. The obtained results unify, extend, and generalize some comparable results in the existing literature.

A generalization of a viscosity generalized Halpern iteration scheme is analyzed. It is proven that the solution converges asymptotically strongly to a unique fixed point of an asymptotically nonexpansive mapping which drives the iteration together with a contractive self-mapping, a viscosity term and two driving external forcing terms.

A generalization of Halpern's iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern's iteration, a contractive mapping, and a nonexpansive one. The sum of the real coefficient sequences of four of the above terms is not required to be unity at each sample but it is assumed to converge asymptotically to unity. Halpern's iteration solution is proven to converge strongly to a unique fixed point of the asymptotically nonexpansive mapping.

This paper discusses a more general contractive condition for a class of extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are nonempty, closed and convex, then all the iterations converge to a unique closed limiting finite sequence, which contains the best proximity points of adjacent subsets, and reduce to a unique fixed point if all such subsets intersect.

Some mutual relations between p-cyclic contractive self-mappings, p-cyclic Kannan self-mappings, and Meir-Keeler p-cyclic contractions are stated. On the other hand, related results about the existence of the best proximity points and existence and uniqueness of fixed points are also formulated.

This paper investigates some properties of cyclic fuzzy maps in metric spaces. The convergence of distances as well as that of sequences being generated as iterates defined by a class of contractive cyclic fuzzy mapping to fuzzy best proximity points of (non-necessarily intersecting adjacent subsets) of the cyclic disposal is studied. An extension is given for the case when the images of the points of a class of contractive cyclic fuzzy mappings restricted to a particular subset of the cyclic disposal are allowed to lie either in the same subset or in its next adjacent one.

Jack the educator. The Syracuse Mathematics Department is housed in Carnegy Library. Imagine heavy snowfall and a slim, tall figure approaching the Carnegy building, propping the heavy door open with one knee while his hands are busy shaking the snow off the Daily Orange (SU student newspaper) and the figure seemingly freezing in this position for as long as it takes to read the front page article. For SU students this is a familiar image of Jack E. Graver, always interested in students and always finding time to read their news. Jack the administrator. A surprisingly deep dark voice resonating from the chairman's office inside Carnegie singing "Nobody knows the trouble I have seen, nobody knows my sorrows" is another image of Jack E. Graver. Jack the researcher. The mathematician Graver is described on MathSciNet by publication in the areas of biology and other natural sciences, combinatorics, convex and discrete geometry, game theory, economics, social and behavioral sciences, geometry manifolds and cell complexes, operations research, mathematical programming, probability theory and stochastic processes. Graver received his Ph.D. from Indiana University in 1964 under Andrew Hugh Wallace. His dissertation's title was "An Analytic Triangulation of an Arbitrary Real Analytic Variety", the field algebraic topology. It is remarkable that the 1964 paper [3] of the same title is cited more than once in this millennium. In 1966 he cashed an Erdös check for results in [8], a paper that may be called his first major work (cited more than 10 times in this millennium!). Because of this paper, written with Jim Yackel at Dartmouth ( John Wesley Young Research Instructor), Jack Graver is known as a Ramsey Theorist. From 1966 to present, he teaches at Syracuse University. Why Syracuse? Because I felt at home here, he says. In 1975 he published On the foundations of linear and integer linear programming I [4], a paper that turned his name into an adjective. On Wickipedia you can read up on Graver bases, but if you want a more reliable source, try , where the relationship of Gröbner bases to Hilbert bases and Graver bases is presented. If you look for On the foundations of linear and integer linear programming II, you need to read Amir Fourudi's thesis, one of Jack Graver's 9 (so far) Ph.D. students -Jack himself was handicapped by his 1977-1994 chairmanship. However, despite administrative duties, his collaboration with Mark Watkins turned him to Graph Theory, as major contributions to the field. A fortunate diversion into Architecture produced not only a novel and unusual text [1], but awoke Jack's interest in rigidity of frameworks and he promoted matroids as a major tool in a colloquium talk that changed my life. I became his Ph.D. student, one of four simultaneously supervised by an acting chair. We did not fight over mathematical issues as Jack had all four of us work on totally disjoint topics, we merely fought for being next in line to enter the chairman's office. Combinatorial rigidity [2] (100 citations) is to this very day the focus of my research. xix Meanwhile Jack has moved on to new Ph.D. students and new topics, namely, Fullerenes . When asked about retirement he justifies the non-existence of plans by stating with a smile: "Normally people retire around 65, but next year will be only my 50 th year at SU, so I have some time to think about that." There's more to life than math-there is family, scouting, Shakespeare, gardening, fine dining, stories. Whenever you tell him a story, Jack will tell you a funnier and better one than you have ever heard. We have told but a small part of an interesting life to which we may look up to for future inspiration.

Jack the educator. The Syracuse Mathematics Department is housed in Carnegy Library. Imagine heavy snowfall and a slim, tall figure approaching the Carnegy building, propping the heavy door open with one knee while his hands are busy shaking the snow off the Daily Orange (SU student newspaper) and the figure seemingly freezing in this position for as long as it takes to read the front page article. For SU students this is a familiar image of Jack E. Graver, always interested in students and always finding time to read their news.

Abstract: We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is ...

We introduce Chronon Field Theory (CFT) as a minimal and self-consistent framework in which conventional spacetime geometry and gauge interactions emerge from a single dynamical temporal field, eliminating the need for a fundamental spacetime manifold. Specifically, a smooth, unit-norm, future-directed timelike vector field Φ µ (x) generates local causal structure, effective spacetime geometry, and emergent gauge interactions without reliance on a background metric or fundamental gauge bosons. The theory yields Einstein-like gravitational dynamics, with the effective metric and connection derived from the foliation induced by Φ µ. A local U (1) symmetry arises from internal phase rotations, giving rise to Maxwell electrodynamics in the weak-field limit, with the photon appearing as a Goldstone-like excitation. We derive explicit field equations, demonstrate local energy conservation from Noether currents, and identify vacuum birefringence as a concrete observational signature potentially accessible to forthcoming CMB polarization experiments. If confirmed, we suggest that CFT could support practical technologies currently limited by relativistic time ambiguity-such as a globally synchronized, GPS-independent positioning and communication network based on direct coupling to the underlying temporal field. CFT thereby provides a unified and empirically accessible alternative to standard geometric and gauge-theoretic foundations of fundamental physics.

We present Chronon Field Theory (CFT), a covariant framework in which large-scale spacetime geometry-including temporal order and causal structure-emerges from the coarse-grained dynamics of a single future-directed timelike field Φ µ (x). Within this approach, both the unit-norm property of Φ µ and the Lorentzian signature of g µν are not postulated but arise uniquely from random microscopic chronon dynamics under broad symmetry and locality assumptions. Quantum phenomena are reinterpreted in this geometric setting: double-slit interference results from phase-coherent Φ-threads rather than superposed particle paths, and measurement appears as a boundary-induced alignment of microscopic orientations with the apparatus's macroscopic field, yielding definite outcomes without collapse or branching. Schrödinger's cat is then a straightforward macro-micro coupling with no persistent macroscopic superposition, while EPR-type entanglement reflects shared Φ-thread ancestry across spacelike separations. The Born rule follows from the unbiased martingale structure of alignment dynamics and from coarse-graining symmetries, making it an emergent statistical law; the uncertainty principle appears as a statistical bound on chronon fluctuations, with h arising as the variance of coarse-grained action. CFT thus provides a covariant and ontologically economical framework linking spacetime symmetries to quantum measurement, and predicts scale-dependent effects-such as nonlinear decoherence and curvature-sensitive timing shifts-that are open to experimental test.

We present a self-contained formulation of Chronon Field Theory (CFT) in which (i) a smooth, unit-norm, future-directed timelike field Φ µ induces foliation, causal structure, and an emergent Lorentzian metric; (ii) a covariant local mass/energy density is defined as ρ(x) = T µν Φ µ Φ µ , furnishing a unified and positive notion of inertial/gravitational mass; (iii) matter arises as topologically stable solitons with w = 1, carrying spin-1 2 and Fermi-Dirac statistics via a Finkelstein-Rubinstein/Berry holonomy mechanism; and (iv) a U(1) gauge sector emerges from chronon holonomy, yielding Maxwell dynamics on the emergent metric and a massless "photon" as a Goldstone-like excitation. We prove positivity and conservation properties for ρ, establish existence of finite-energy soliton minimizers under mild assumptions, and state a holonomy-matching theorem that ties the FR Z 2 class to the emergent spacetime U(1) connection. A key outcome is that the familiar constants of Nature are not postulated but emerge: the effective action unit ℏ eff ∼ E c τ c , Newton's constant G ∼ (c(α i , γ)Λ 2)-1 , the elementary charge e = q 0 / √ κ A , and the universal light speed c = ℓ Φ /τ c. In the gravitational sector we recover Einstein-Maxwell on stabilized domains at two-derivative order, while allowing controlled, power-counted deviations (aether-like and higher-derivative terms) that are constrained phenomenologically. We further derive the Coulomb law on stabilized leaves, V (r) = e 1 e 2 /(4πr), whose normalization fixes the holonomy stiffness via the observed Coulomb constant (equivalently α em), yielding κ A = q 2 0 /(4πα em). We outline collective-coordinate quantization where splittings scale with ℏ eff , and identify falsifiable signatures (achromatic birefringence; exchange-phase interferometry). This paper (I) lays the foundation for non-Abelian extensions (II) and QCD-like dynamics (III).

We extend Chronon Field Theory (CFT) beyond the emergent U(1) sector by constructing a compact non-Abelian holonomy from the internal geometry of the chronon field Φ µ. We prove existence of an emergent principal SU(2) bundle and connection, derive the Yang-Mills action on the emergent metric g µν [Φ], and show how vector boson masses arise without a fundamental scalar: either via a Stückelberg-like realization of internal phases or through a composite amplitude mode of Φ. We demonstrate that a residual U(1) combination remains unbroken, ensuring an exactly massless photon as in QED, and we establish Ward/Slavnov-Taylor identities and tree-level unitarity for longitudinal vector scattering. Solitonic matter from Paper I is minimally coupled through the emergent connection; we extract phenomenological constraints from precision electroweak data and outline a path to SU(3) (Paper III).

not derived from nor constructed using Euler's constant e, nor the golden ratio φ. Their origin is completely independent, based on new rational structures that model energy flows on differentiable surfaces. This framework represents a breakthrough in the formalization of dynamic particle and energy flows from a geometric and rational perspective, with potential applications in mathematical physics, field theory, and differential geometry.

We re-derive Thales, Pythagoras, Apollonius, Stewart, Heron, al Kashi, de Gua, Terquem, Ptolemy, Brahmagupta and Euler's theorems as well as the inscribed angle theorem, the law of sines, the circumradius, inradius and some angle bisector formulae, by assuming the existence of an unknown relation between the geometric quantities at stake, observing how the relation behaves under small deviations of those quantities, and naturally establishing differential equations that we integrate out. Applying the general solution to some specific situation gives a particular solution corresponding to the expected theorem. We also establish an equivalence between a polynomial equation and a set of partial differential equations. We finally comment on a differential equation which arises after a small scale transformation and should concern all relations between metric quantities.

We re-derive Thales, Pythagoras, Apollonius, Stewart, Heron, al Kashi, de Gua, Terquem, Ptolemy, Brahmagupta and Euler's theorems as well as the inscribed angle theorem, the law of sines, the circumradius, inradius and some angle bisector formulae, by assuming the existence of an unknown relation between the geometric quantities at stake, observing how the relation behaves under small deviations of those quantities, and naturally establishing differential equations that we integrate out. Applying the general solution to some specific situation gives a particular solution corresponding to the expected theorem. We also establish an equivalence between a polynomial equation and a set of partial differential equations. We finally comment on a differential equation which arises after a small scale transformation and should concern all relations between metric quantities.

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We establish a bijective correspondence between affine connections and a class of semi-holonomic jets of local diffeomorphisms of the underlying manifold called symmetry jets in the text. The symmetry jet corresponding to a torsion free connection consists in the family of 2-jets of the geodesic symmetries. Conversely, any connection is described in terms of the geodesic symmetries by a simple formula involving only the Lie bracket of vector fields. We then formulate, in terms of the symmetry jet, several aspects of the theory of affine connections and obtain geometric and intrinsic descriptions of various related objects involving the gauge groupoid of the frame bundle. In particular, the property of uniqueness of affine extension admits an equivalent formulation as the property of existence and uniqueness of a certain groupoid morphism. Moreover, affine extension may be carried out at all orders and this allows for a description of the tensors associated to an affine connections, namely the torsion, the curvature and their covariant derivatives of all orders, as obstructions for the affine extension to be holonomic. In addition this framework provides a nice interpretation for the absence of other tensors. Contents 2 Connections with torsion 3 Uniqueness of affine extension revisited 4 Torsion and holonomy of affine jets 5 The curvature in terms of the symmetry jet 6 Third order affine extension 7 The curvature as a measure of the integrability of affine jets 8 Why there are no other tensors than the torsion and the curvature ? 9 Distributions on associated bundles 10 Levi-Civita connection 11 Relation with Lie algebroid connections 12 Geodesics and parallel transport 13 Geodesic symmetries and locally symmetric spaces 14 Relation with Kobayashi's admissible sections A Notations B Lie groupoids of jets of bisections C Second tangent bundle D Structure of B (1,1) (Pair(M )) E Third tangent bundle F Structure of B (1,1,1) nh (Pair(M )