Biconservative and biharmonic surfaces in Euclid and Minkowski spaces

dc.contributor.advisor Turgay, Nurettin Cenk
dc.contributor.advisor Şen Yeğin, Rüya
dc.contributor.author Yürük, Hazal
dc.contributor.authorID 509211217
dc.contributor.department Mathematics Engineering
dc.date.accessioned 2025-03-25T06:17:24Z
dc.date.available 2025-03-25T06:17:24Z
dc.date.issued 2024-07-02
dc.description Thesis (M.Sc.) -- İstanbul Technical University, Graduate School, 2024
dc.description.abstract In 1964, Sampson and Eells formulated the concept of biharmonic maps as an extension of harmonic maps while investigating the energy functional $E$ between Riemannian manifolds, a subject of both geometric and physical significance. Subsequently, numerous mathematicians have shown interest in the study of biharmonic mappings.\\ By the definition, the bienergy functional between semi-Riemannian manifolds $(M^m,g)$ and $(N^n,\tilde{g})$ is defined by $$E_2(\varphi)=\frac {1}{2} \int_M \|\tau(\varphi)\|^2 v_g$$ for a smooth map $\varphi:M \to N$, where $\tau(\varphi)$ represents the tension field of $\varphi$. $\varphi:M\to N$ is said to be biconservative if it is a critical point of $E_2$. This condition is equivalent to satisfying the Euler-Lagrange equation associated with the bienergy functional $$\tau_2(\varphi)=0,$$ where $\tau_2$ is the bitension field defined by $$\tau_2(\varphi):=\Delta\tau(\varphi)-\mathrm{tr\,} \tilde{R}(d\varphi,\tau(\varphi))d\varphi.$$\\ In the 1980s, B. Y. Chen conducted research on biharmonic submanifolds within Euclidean spaces as a component of B. Y. Chen's initiative to comprehend submanifolds of finite type in semi-Euclidean spaces. B. Y. Chen proposed an other characterization of biharmonic submanifolds in these spaces. Let $x:M\to \mathbb E^n_r$ be an isometric immersion. By examining normal and tangential parts of $\tau_2(x)$, the following results can be obtained.\\ Let $x: M^m \rightarrow \mathbb E^n_r$ be an isometric immersion of an $n$-dimensional semi-Riemannian submanifold $M^m$ into the semi-Euclidean space $\mathbb E^n_r$. If $x$ satisfies the fourth-order semi-linear PDE system given by the equations $$\Delta^\perp H+\mathrm{trace}h(A_H(\cdot),\cdot)=0$$ and $$m\mathrm{grad} \Vert H \Vert^2 +4\mathrm{trace} A_{\nabla^\perp_\cdot H}(\cdot)=0,$$ then $M^m$ is biharmonic.\\ On the other hand, if a mapping $\varphi: M \to N$ satisfies the weaker condition $$\langle \tau_2(\varphi), d\varphi \rangle = 0,$$ then it is said to be biconservative. Mainly, if $x: M \to N$ is an isometric immersion, then the previous equation is equivalent to $$\tau_2(x)^\top = 0,$$ where $\tau_2(x)^\top$ represents the tangential part of $\tau_2(x)$. In this case, $M$ is said to be a biconservative submanifold of $N$.\\ In this thesis, we mainly focus on biharmonic and biconservative surfaces in four dimensional Euclidean and Minkowski spaces. The first section provides a concise overview of the historical background and underlying principles concerning biharmonic and biconservative submanifolds, as well as an overview of the research conducted far in this field. In the second section, we give some basic notations and basic facts about submanifolds of semi- Euclidean spaces, the definition of biconservative submanifolds and we introduce the rotational surfaces. In the third section, we give biconservative PNMCV surfaces in $\mathbb E^4$. We obtain local parameterizations of these surfaces and demonstrate that they are not biharmonic. In the fourth section, we give biconservative rotational surfaces in $\mathbb E_1^4$. We study with three different class of rotational surfaces and obtain the condition for each of them to be biconservative. In the concluding section, the derived conclusions are presented, along with recommendations regarding possible future researches.
dc.description.degree M.Sc.
dc.identifier.uri http://hdl.handle.net/11527/26673
dc.language.iso en_US
dc.publisher Graduate School
dc.sdg.type Goal 3: Good Health and Well-being
dc.sdg.type Goal 4: Quality Education
dc.subject Biconservative Submanifolds
dc.subject Çift Konservatif Altmanifoldlar
dc.subject Biharmonic surfaces
dc.subject Biharmonik yüzeyler
dc.subject Biconservative
dc.subject Bikonzörvatif
dc.title Biconservative and biharmonic surfaces in Euclid and Minkowski spaces
dc.title.alternative Öklid ve Minkowski uzaylarındaki bikonzörvatif ve biharmonik yüzeyler
dc.type Master Thesis
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