Introduction to Differential Geometry in the Three-Dimensional Field of Complex Vectors

[2608.14686] Introduction to Differential Geometry in the Three-Dimensional Field of Complex Vectors Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:2608.14686 (math) [Submitted on 5 Aug 2026] Title: Introduction to Differential Geometry in the Three-Dimensional Field of Complex Vectors Authors: Branko Sarić View a PDF of the paper titled Introduction to Differential Geometry in the Three-Dimensional Field of Complex Vectors, by Branko Sari\'c View PDF HTML (experimental) Abstract: Building on a commutative geometric algebra developed for 2D and 3D fields of complex vectors, this paper develops a vector framework for the differential geometry of compact Hermitian manifolds embedded in an ambient three-dimensional field of complex vectors. The underlying algebraic framework is constructed from the commutative geometric product of complex vectors, yielding a unified commutative algebra in both the 2D and 3D settings together with the corresponding vector differential and integral identities. These algebraic results provide the intrinsic foundation for a differential geometric formulation that is developed independently of the classical scalar coordinate approach. Within this framework, compact Hermitian manifolds are described entirely in vector form. Fundamental tensor relations governing the intrinsic and extrinsic geometry of complex manifolds are derived in vector form. The resulting framework establishes a unified algebraic and geometric setting in which the differential geometry of complex manifolds is developed directly from the commutative algebra of complex vectors. Comments: 16 pages Subjects: Differential Geometry (math.DG) MSC classes: Primary 53A45, 14A05, Secondary 32Q35, 15A75 ACM classes: G.1.9 Cite as: arXiv:2608.14686 [math.DG] (or arXiv:2608.14686v1 [math.DG] for this version) https://doi.org/10.48550/arXiv.2608.14686 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Branko Sarić [ view email ] [v1] Wed, 5 Aug 2026 12:15:27 UTC (29 KB) Full-text links: Access Paper: View a PDF of the paper titled Introduction to Differential Geometry in the Three-Dimensional Field of Complex Vectors, by Branko Sari\'c View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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