Metric <i>f</i>-Contact Manifolds Satisfying the (<i>κ</i>, <i>μ</i>)-Nullity Condition
We prove that if the <i>f</i>-sectional curvature at any point of a <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mi>s</mi> <mo>)...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/6/891 |
| _version_ | 1797566362296516608 |
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| author | Alfonso Carriazo Luis M. Fernández Eugenia Loiudice |
| author_facet | Alfonso Carriazo Luis M. Fernández Eugenia Loiudice |
| author_sort | Alfonso Carriazo |
| collection | DOAJ |
| description | We prove that if the <i>f</i>-sectional curvature at any point of a <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mi>s</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-dimensional metric <i>f</i>-contact manifold satisfying the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>κ</mi> <mo>,</mo> <mi>μ</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> nullity condition with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> is independent of the <i>f</i>-section at the point, then it is constant on the manifold. Moreover, we also prove that a non-normal metric <i>f</i>-contact manifold satisfying the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>κ</mi> <mo>,</mo> <mi>μ</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> nullity condition is of constant <i>f</i>-sectional curvature if and only if <inline-formula> <math display="inline"> <semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mi>κ</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and we give an explicit expression for the curvature tensor field in such a case. Finally, we present some examples. |
| first_indexed | 2024-03-10T19:25:53Z |
| format | Article |
| id | doaj.art-5e192fb3fcd74d3ca67e6f6334a08b38 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T19:25:53Z |
| publishDate | 2020-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-5e192fb3fcd74d3ca67e6f6334a08b382023-11-20T02:33:55ZengMDPI AGMathematics2227-73902020-06-018689110.3390/math8060891Metric <i>f</i>-Contact Manifolds Satisfying the (<i>κ</i>, <i>μ</i>)-Nullity ConditionAlfonso Carriazo0Luis M. Fernández1Eugenia Loiudice2Departamento de Geometría y Topología, c/Tarfia s/n, Universidad de Sevilla, 41012 Sevilla, SpainDepartamento de Geometría y Topología, c/Tarfia s/n, Universidad de Sevilla, 41012 Sevilla, SpainFachbereich Mathematik und Informatik, Philipps Universität Marburg, Hans-Meerwein-Straße, 35032 Marburg, GermanyWe prove that if the <i>f</i>-sectional curvature at any point of a <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mi>s</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-dimensional metric <i>f</i>-contact manifold satisfying the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>κ</mi> <mo>,</mo> <mi>μ</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> nullity condition with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> is independent of the <i>f</i>-section at the point, then it is constant on the manifold. Moreover, we also prove that a non-normal metric <i>f</i>-contact manifold satisfying the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>κ</mi> <mo>,</mo> <mi>μ</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> nullity condition is of constant <i>f</i>-sectional curvature if and only if <inline-formula> <math display="inline"> <semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mi>κ</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and we give an explicit expression for the curvature tensor field in such a case. Finally, we present some examples.https://www.mdpi.com/2227-7390/8/6/891metric <i>f</i>-contact manifold<i>f</i>-(<i>κ</i>,<i>μ</i>) manifold<i>f</i>-(<i>κ</i>,<i>μ</i>)-space form |
| spellingShingle | Alfonso Carriazo Luis M. Fernández Eugenia Loiudice Metric <i>f</i>-Contact Manifolds Satisfying the (<i>κ</i>, <i>μ</i>)-Nullity Condition Mathematics metric <i>f</i>-contact manifold <i>f</i>-(<i>κ</i>,<i>μ</i>) manifold <i>f</i>-(<i>κ</i>,<i>μ</i>)-space form |
| title | Metric <i>f</i>-Contact Manifolds Satisfying the (<i>κ</i>, <i>μ</i>)-Nullity Condition |
| title_full | Metric <i>f</i>-Contact Manifolds Satisfying the (<i>κ</i>, <i>μ</i>)-Nullity Condition |
| title_fullStr | Metric <i>f</i>-Contact Manifolds Satisfying the (<i>κ</i>, <i>μ</i>)-Nullity Condition |
| title_full_unstemmed | Metric <i>f</i>-Contact Manifolds Satisfying the (<i>κ</i>, <i>μ</i>)-Nullity Condition |
| title_short | Metric <i>f</i>-Contact Manifolds Satisfying the (<i>κ</i>, <i>μ</i>)-Nullity Condition |
| title_sort | metric i f i contact manifolds satisfying the i κ i i μ i nullity condition |
| topic | metric <i>f</i>-contact manifold <i>f</i>-(<i>κ</i>,<i>μ</i>) manifold <i>f</i>-(<i>κ</i>,<i>μ</i>)-space form |
| url | https://www.mdpi.com/2227-7390/8/6/891 |
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