On the other hand, the geometry of submanifolds on Riemannian manifolds is an important topic of research in differential geometry. The purpose of the special issue "Differential Geometry" of the journal Matematika was to provide a collection of papers that reflect modern research topics and new developments in the field of differential geometry and explore applications in other fields.
The Characterization of Affine Symplectic Curves in R 4
Introduction
Finally, in section 4, we present the results we obtain on the characterizations of the asymptotic curves in R4 and the study of the asymptotic helices.
Preliminaries
Let z:I→R4 be a sympectic regular curve, which is parameterized by the sympectic arc length, and such that H2(s) =0. Let z: I→R4 be a sympectic regular curve, parameterized by the sympectic arc length, and {a1(s),a2(s),a3(s),a4(s)}be the Frenet frame of z .
Conclusions
The theorem of Cartan states that the curves with constant symplectic curvatures are exactly the trajectories of the one-parameter subgroups of the affine symplectic group in four variables [23,24]. Depending on the two cases involving sympectic curvatures, we get sympectic general helixes of the Euclidean or hyperbolic type.
Euclidean Submanifolds via Tangential Components of Their Position Vector Fields
- Rectifying Submanifolds of Riemannian Manifolds
- Euclidean Submanifolds with x T as Potential Fields
- Interactions between Torqued Vector Fields and Ricci Solitons
- Conclusions
If the gradient ∇f of f is a torsion vector field, then it is a concircular vector field on M. If a torsion vector field on a Riemannian manifold M is a gradient vector field, then it is a concircular vector field.
A New Proof of a Conjecture on Nonpositive Ricci Curved Compact Kähler–Einstein Surfaces
- Existence of Ball-Like Points
- Generalized Hong-Cang Yang’s Function
- The Generalization
- Appendix
The second theorem also follows from the argument from [1] by applying it to the maximum direction instead of the minimum direction. It is clear that in the case of the maximal direction we must assume that B∗ =0.
Comparison of Differential Operators with Lie
Derivative of Three-Dimensional Real Hypersurfaces in Non-Flat Complex Space Forms
Proofs of Theorems 1 and 2
Assume that M is a real hypersurface in M2(c) whose shape operator satisfies relation (5), which due to the relation between k-th generalized Tanaka-Webster connection (1) becomes. Thus λ = ν = α and the real hypersurface is completely umbilical, which is a contradiction, and this completes the proof of Theorem1.
Proof of Theorems 4 and 5
A real hypersurface M in M2(c) whose tensor field P satisfies relation (7) is locally congruent with a real hypersurface of type (A). In this paper, we answer the question whether there are real three-dimensional hypersurfaces in non-flat complex space forms whose differential operator L(k) of a tensor field of type (1, 1) coincides with its Lie derivative. The obtained results complement the work that has been done in the case of real hypersurfaces with dimensions greater than three in the complex projective space (see [11]).
On the Lie derivative of real hypersurfaces in CP2 and CH2 with respect to the generalized Tanaka-Webster connection. Bull.
Inequalities on Sasakian Statistical Manifolds in Terms of Casorati Curvatures
Inequalities with Casorati Curvatures
Some geometrical comments on vision and neurobiology: Seeing Gauss and Gabor pass by, looking through the window of Parma in Leuven in the company of Casorati. Optimal inequalities for the Casorati curvatures of submanifolds of real space shapes equipped with semisymmetric metric connections.J. Optimal inequalities for the Casorati curvatures of submanifolds of generalized spatial forms equipped with semisymmetric metric connections. Bull.
Inequalities for generalized δ-Casorati curvatures of submanifolds in real space forms equipped with semisymmetric metric connection. Rev.
Pinching Theorems for a Vanishing C-Bochner Curvature Tensor
Inequalities Involving a Vanishing C-Bochner Curvature Tensor
When a submanifoldMis Einstein of a Sasakian manifold(M,g,ϕ,ξ,η), the Ricci curvature tensorρ(X,Y) = λg(X,Y)forX,Y ∈ Γ(TM), whereλ is a constant. Furthermore, the equality case holds if and only if Mnis an invariant quasi-umbilical submanifold with the trivial normal connection in a Sasakian manifold(M,g,ϕ,ξ,η), such that with respect to a suitable orthonormal tangent frame{ξ1 ,· · · ,ξn}and a normal orthonormal frame{ξn+1,· · ·,ξm}, the form operators Ar≡Aξr and r∈ {n+1,· · ·,m} have the form of equation( 8). For an oblique submanifold(g(ϕei,ej) = cosθwith the oblique angleθ) of a Sasakian manifold (M,g,ϕ,ξ,η) with a vanishing C-Bochner curvature tensor, we have the following consequences.
Optimal inequalities for the Casorati curvatures of submanifolds in generalized space forms equipped with semi-symmetric non-metric connections.Symmetrie2016,8, 10.
Curvature Invariants for Statistical Submanifolds of Hessian Manifolds of Constant Hessian Curvature
Statistical Manifolds and Their Submanifolds
Using the Hessian curvature tensor ˜Q, a Hessian sectional curvature can be defined on a Hessian manifold. XY=∇∗XY+h∗(X,Y), (6) whereh,h∗:Γ(TMn)×Γ(TMn)→Γ(T⊥Mn)are symmetric and bilinear, called the embedded curvature tensor of Mnin ˜ Mmfor ˜∇and the embedded curvature tensorofMnin ˜Mmfor ˜∇∗, respectively. 10] Let∇˜ and∇˜∗be double connections on a statistical manifoldM˜mand∇the induced connection by∇˜ on a statistical submanifold Mn. LetR and R be the Riemannian curvature tensors for˜∇˜ and.
10] Let the connections ∇˜ and∇˜∗ be dual connections on a statistical manifold M˜mand∇∗ the connection induced by∇˜∗on a statistical submanifold Mn. ∗, respectively.
Euler Inequality and Chen-Ricci Inequality
For the equations of Gauss, Codazzi and Ricci regarding the compound ˜∇∗opMn we have Theorem 2. We then establish a Chen-Ricci inequality for statistical submanifolds in Hessian manifolds with constant Hessian curvature. Imply Gauss equation for the Levi-Civita connection and the definition of the Hessian cross-sectional curvature.
But the Ricci curvature R0 with respect to the Levi-Civita connection is given by 2Ric0(X) =2τ0−.
Completness of Statistical Structures
Statistical Structures with Complete Metrics
If the cross-sectional ∇-curvature is everywhere non-negative, then the statistical structure is trivial, i.e. ∇=∇. If the statistical sectional curvature is bounded from ˆ 0 by a positive constant, then furthermore M is compact and its first fundamental group finite. Moreover, the statistical connection on the affine sphere is projectively flat and its ∇-section curvature is constant.
Furthermore, if the diameterˆ ∇ curvature of0 is bounded by a positive constant, then M is compact and its first fundamental group is finite.
Completeness of Statistical Connections
In particular, every function on a compact Riemannian manifold gives rise to a statistical structure on which the statistical connection is complete. Every function σ on M results in a statistical structure whose statistical connection and its conjugate are complete. Lauritzen S.L. Statistical collectors; IMS Lecture Notes-Monograph Series 10; Institute for Mathematical Statistics: Beachwood, OH, USA, 1987; p.
L 2 -Harmonic Forms on Incomplete Riemannian Manifolds with Positive Ricci Curvature †
Warped Product Manifolds
We consider L2 harmonic forms, the Ricci curvature and the capacity of the Cauchy limit for a general distorted product Riemannian manifold. Assume that f(r) is the same order afor some constant 0
Now, we calculate the capacity of the Cauchy limit∂cM:=M\M={x0}, whereMis the completion as the metric spaceM with respect to the Riemannian distancedg.
The Proof of Theorem 2
By means of the partition of the unit it is sufficient to put the properties(1) through(4) in Theorem2 on one side hornCsina(r)(Tn)reg= ((0,π2)×Tn,dr2⊕ sin2a( r)h). Thus, the author thinks it would be important to study incomplete Riemann manifolds for which the L2-Stokes theorem does not hold. Cheeger, J.; Goresky, M.; MacPherson, R.L2 Cohomology and Intersection Homology of Simple Algebraic Varieties, Seminar on Differential Geometry.Ann.
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
On Angles and Pseudo-Angles in Minkowskian Planes
- The Pseudo-Angles of Helzer
- The Minkowskian Pseudo-Angles between A Null Direction and Any Spacelike or Timelike Direction
- The Minkowskian Angles between Null Directions
- The Unoriented Minkowskian Angles and Pseudo-Angles
- A Geometrical Meaning of the Minkowskian Angles and Pseudo-Angles
- Conclusions
Moreover, the main goal of this paper is to properly define Minkowski angles and pseudo-angles between the two zero directions and between a zero direction and any non-zero direction, respectively. Moreover, the oriented Minkowski angles or pseudo-angles between these directions then correspond to the oriented arc lengths on H and the oriented pseudo lengths of parts of the zero lines D1 and D2, these pseudo lengths coming about in an oriented manner as suggested. in Figure 10. This paper gives a geometrical generalization of the Euclidean angle measure between two directions as the Euclidean lengths of corresponding arcs on a Euclidean unit circle for two directions with arbitrary causal characters in a Minkowski plane, by a well-defined notion of the Minkowski angles or pseudo angles of these two directions.
And, of course, the classical Minkowskian angles between any two spatial directions and between any two time-like directions within the same branch of the Minkowskian unit circle fit properly with the above-given notion of central angles and Minkowskian pseudo-angles.
Generic Properties of Framed Rectifying Curves
- Framed Curve and Adapted Frame
- Framed Rectifying Curves
- Framed Rectifying Curves versus Centrodes
- Contact between Framed Rectifying Curves
The centrode of a boxed curve with a non-zero constant boxed-curvature function p(s) and a non-constant boxed-curvature function q(s) is a boxed rectifier curve. Conversely, the boxed rectifier curve in R3 is the centrode of a given boxed curve with a non-constant boxed curvature function q(s) and a non-zero constant boxed curvature function p(s). Therefore, the boxed rectifier curve γ is the centrode of a boxed curve with non-constant boxed curvature q(s) and non-zero constant boxed curvaturep.
The center of a framed curve with a non-zero constant framed curved function q(s) and a non-constant framed curved function p(s) is a framed rectification curve.
Hypersurfaces with Generalized 1-Type Gauss Maps
Surfaces with Generalized 1-Type Gauss Maps
Suppose that a developable surface in E3 has a generalized 1-type Gaussian map, that is, the Gaussian map of the surface satisfies the condition. It follows from (10) that M has generalized 1-type Gaussian map with C=0, i.e. Mha is a pointwise 1-type Gaussian map of the first kind if and only if M has a constant mean curvature, H. Suppose that M has a generalized 1-type Gaussian map, that is, the Gaussian map, G, of the conical surface satisfies (11).
A tangent surface developed in E3 with a generalized Gaussian type 1 map is an open part of a plane.
Cylindrical Hypersurfaces with Generalized 1-Type Gauss Maps
Thus, by summarizing all the results in this section, we established the following classification theorem for developable surfaces with generalized 1-type Gaussian maps: To find the best possible estimate of the total mean curvature of a compact submanifold of Euclidean space, Chen introduced the study of finite type submanifolds. Space-like rotational surfaces of elliptic, hyperbolic and parabolic types in Minkowski spaceE41 with pointwise 1-type Gauss map.Mathematics.
Inextensible Flows of Curves on Lightlike Surfaces
- Inextensible Flows of a Spacelike Curve
- Inextensible Flows of a Lightlike Curve
- Lightlike Ruled Surfaces
- Conclusions
Let M be the light-like surface in the Minkowski R31 three-space and let {t,n,g} be the Darboux frames along the light-like curve γ on M. Then we give the time evolution equations of the Darboux frame of the light-like curve on the light-like surface. Let X(u,v,t) = α(u,t) +vα(u,t) be the surface evolution of the light-like tangent developed surface given by (24)v R31, and ∂α∂t = f1t+f2n+ f3g , where,n,arrange the Darboux frames along the light-like curveα on the light-like surface.
We study an inextensible flow of a space-like or light-like curve on a light-like surface in Minkowski three-space and investigate a time evolution of the Darboux frame{t,n,g}(see Theorems 3 and 7) and the functionsκn,κgandτg(see Theorems 4 and 8) .
Trans-Sasakian 3-Manifolds with Reeb Flow Invariant Ricci Operator
Trans-Sasakian Manifolds
In particular, a three-dimensional almost contact metric manifold is trans-Sasakian if and only if it is normal (see [24,25]). According to [16], in a locally symmetric trans-Sasakian 3-manifold, the Reeb vector field is a vector field of the Ricci operator. As seen in the proof of Theorem 1, a 3 trans-Sasakian manifold with Ricci invariant Reeb flow operator is an aα-Sasakian manifold, a cosymplectic manifold or a space with constant sectional curvature.
It was proved in [17] (Lemma 5.1) that when ξ of a compact trans-Sasakian 3-manifold is minimal or harmonic, then α is a constant.
