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# The Hopf fibration 4

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

We recall some classical results on the Hopf fibration $f:S^3 \rightarrow S^2$. We focus on the preimage of a curve gamma on $S^2$ via the projection $f$. It is known as the Hopf tube over gamma and we give some curvature properties. We point out, as application related to Physics, some developments on magnetic curves on the 3-dimensional sphere. We complete the lectures extending all these studies to the fibration $M^3(c) \rightarrow S^2(r)$, where $M^3(c)$ is an elliptic Sasakian space form.

# The Hopf fibration 3

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

We recall some classical results on the Hopf fibration $f:S^3 \rightarrow S^2$. We focus on the preimage of a curve gamma on $S^2$ via the projection $f$. It is known as the Hopf tube over gamma and we give some curvature properties. We point out, as application related to Physics, some developments on magnetic curves on the 3-dimensional sphere. We complete the lectures extending all these studies to the fibration $M^3(c) \rightarrow S^2(r)$, where $M^3(c)$ is an elliptic Sasakian space form.

# The Hopf fibration 2

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

We recall some classical results on the Hopf fibration $f:S^3 \rightarrow S^2$. We focus on the preimage of a curve gamma on $S^2$ via the projection $f$. It is known as the Hopf tube over gamma and we give some curvature properties. We point out, as application related to Physics, some developments on magnetic curves on the 3-dimensional sphere. We complete the lectures extending all these studies to the fibration $M^3(c) \rightarrow S^2(r)$, where $M^3(c)$ is an elliptic Sasakian space form.

# The Hopf fibration 1

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

We recall some classical results on the Hopf fibration $f:S^3 \rightarrow S^2$. We focus on the preimage of a curve gamma on $S^2$ via the projection $f$. It is known as the Hopf tube over gamma and we give some curvature properties. We point out, as application related to Physics, some developments on magnetic curves on the 3-dimensional sphere. We complete the lectures extending all these studies to the fibration $M^3(c) \rightarrow S^2(r)$, where $M^3(c)$ is an elliptic Sasakian space form.

# Magnetic maps - V

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

# Magnetic maps - IV

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

# Magnetic maps - III

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

# Magnetic maps - II

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

# Magnetic maps - I

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

# On the geometry of certain surfaces in homogeneous 3-spaces

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

The geometry of surfaces in spaces of dimension 3, especially of the form M(c)^2 x R, has enriched in last years. An interesting problem studied in very recent papers consists of characterization and classification for constant angle surfaces in different 3-dimensional spaces belonging to the Thurston list. Such a surface is an orientable surface whose unit normal makes a constant angle with a fixed direction. When the ambient is of the form M^2 x R, the favored direction is R. It is proved that for a constant angle surface in E^3, S^2 x R or in H^2 x R, the projection of d/dt onto the tangent plane of the immersed surface, denoted by T, is a principal direction with the corresponding principal curvature identically zero. The main topic of this talk is to investigate surfaces in H^2 x R for which T is a principal direction.

Seminario de Matemáticas (1ª Planta, Sección de Matemáticas)

# Marian Ioan Munteanu

## Universitatea Alexandru Ioan Cuza

Number of talks
10
Number of visits
3
Last visit
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Country of origin
Rumanía