Maps generating normals on a manifold
The first- and second-order canonical forms are considered on a smooth m-manifold. The approach connected with coordinate expressions of basic and fiber forms, and first- and second-order tangent vectors on a manifold is implemented. It is shown that the basic first- and second-order tangent ve...
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| Format: | Article |
| Language: | English |
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Immanuel Kant Baltic Federal University
2019-08-01
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| Series: | Дифференциальная геометрия многообразий фигур |
| Subjects: | |
| Online Access: | https://journals.kantiana.ru/geometry/4279/12674/ |
| _version_ | 1818293737524035584 |
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| author | K. Polyakova |
| author_facet | K. Polyakova |
| author_sort | K. Polyakova |
| collection | DOAJ |
| description |
The first- and second-order canonical forms are considered on a smooth m-manifold. The approach connected with coordinate expressions of basic and fiber forms, and first- and second-order tangent vectors on a manifold is implemented. It is shown that the basic first- and second-order tangent vectors are first- and second-order Pfaffian (generalized) differentiation operators of functions set on a manifold. Normals on a manifold are considered.
Differentials of the basic first-order (second-order) tangent vectors are linear maps from the first-order tangent space to the second-order (third-order) tangent space. Differentials of the basic first-order tangent vectors (basic vectors of the first-order normal) set a map which takes a first-order tangent vector into the second-order normal for the first-order tangent space of (into the third-order normal for second-order tangent space).
The map given by differentials of the basic first-order tangent vectors takes all tangent vectors to the vectors of the second-order normal. Such splitting the osculating space in the direct sum of the first-order tangent space and second-order normal defines the simplest (canonical) affine connection which horizontal subspace is the indicated normal. This connection is flat and symmetric. The second-order normal is the horizontal subspace for this connection.
Derivatives of some basic vectors in the direction of other basic vectors are equal to values of differentials of the first vectors on the second vectors, and the order of differentials is equal to the order of vectors along which differentiation is made.
The maps set by differentials of basic vectors of r-order normal for
(r – 1)-order tangent space take the basic first-order tangent vectors to the basic vectors of (r + 1)-order normal for r-order tangent space.
Horizontal vectors for the simplest second-order connection are constructed. Second-order curvature and torsion tensors vanish in this connection. For horizontal vectors of the simplest second-order connection there is the decomposition by means of the basic vectors of the second- and third-order normals. |
| first_indexed | 2024-12-13T03:20:37Z |
| format | Article |
| id | doaj.art-61696c4a62394583b9590efe21fc1ddb |
| institution | Directory Open Access Journal |
| issn | 0321-4796 2782-3229 |
| language | English |
| last_indexed | 2024-12-13T03:20:37Z |
| publishDate | 2019-08-01 |
| publisher | Immanuel Kant Baltic Federal University |
| record_format | Article |
| series | Дифференциальная геометрия многообразий фигур |
| spelling | doaj.art-61696c4a62394583b9590efe21fc1ddb2022-12-22T00:01:22ZengImmanuel Kant Baltic Federal UniversityДифференциальная геометрия многообразий фигур0321-47962782-32292019-08-015011012510.5922/0321-4796-2019-50-13Maps generating normals on a manifoldK. Polyakova0https://orcid.org/0000-0002-9935-0232Immanuel Kant Baltic Federal University The first- and second-order canonical forms are considered on a smooth m-manifold. The approach connected with coordinate expressions of basic and fiber forms, and first- and second-order tangent vectors on a manifold is implemented. It is shown that the basic first- and second-order tangent vectors are first- and second-order Pfaffian (generalized) differentiation operators of functions set on a manifold. Normals on a manifold are considered. Differentials of the basic first-order (second-order) tangent vectors are linear maps from the first-order tangent space to the second-order (third-order) tangent space. Differentials of the basic first-order tangent vectors (basic vectors of the first-order normal) set a map which takes a first-order tangent vector into the second-order normal for the first-order tangent space of (into the third-order normal for second-order tangent space). The map given by differentials of the basic first-order tangent vectors takes all tangent vectors to the vectors of the second-order normal. Such splitting the osculating space in the direct sum of the first-order tangent space and second-order normal defines the simplest (canonical) affine connection which horizontal subspace is the indicated normal. This connection is flat and symmetric. The second-order normal is the horizontal subspace for this connection. Derivatives of some basic vectors in the direction of other basic vectors are equal to values of differentials of the first vectors on the second vectors, and the order of differentials is equal to the order of vectors along which differentiation is made. The maps set by differentials of basic vectors of r-order normal for (r – 1)-order tangent space take the basic first-order tangent vectors to the basic vectors of (r + 1)-order normal for r-order tangent space. Horizontal vectors for the simplest second-order connection are constructed. Second-order curvature and torsion tensors vanish in this connection. For horizontal vectors of the simplest second-order connection there is the decomposition by means of the basic vectors of the second- and third-order normals.https://journals.kantiana.ru/geometry/4279/12674/differential formssecond-order spacepfaffian (generalized) derivativesnormal on manifoldaffine connections |
| spellingShingle | K. Polyakova Maps generating normals on a manifold Дифференциальная геометрия многообразий фигур differential forms second-order space pfaffian (generalized) derivatives normal on manifold affine connections |
| title | Maps generating normals on a manifold |
| title_full | Maps generating normals on a manifold |
| title_fullStr | Maps generating normals on a manifold |
| title_full_unstemmed | Maps generating normals on a manifold |
| title_short | Maps generating normals on a manifold |
| title_sort | maps generating normals on a manifold |
| topic | differential forms second-order space pfaffian (generalized) derivatives normal on manifold affine connections |
| url | https://journals.kantiana.ru/geometry/4279/12674/ |
| work_keys_str_mv | AT kpolyakova mapsgeneratingnormalsonamanifold |